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# Eight Lectures on Theoretical Physics/IV

Fourth Lecture.

The Equation of State for a Monatomic Gas.

My problem today is to utilize the general fundamental laws concerning the concept of irreversibility, which we established in the lecture of yesterday, in the solution of a definite problem: the calculation of the entropy of an ideal monatomic gas in a given state, and the derivation of all its thermodynamic properties. The way in which we have to proceed is prescribed for us by the general definition of entropy:

{\displaystyle {\begin{aligned}&(13){\color {White}.}\qquad &&S=k\log W.\end{aligned}}}

The chief part of our problem is the calculation of ${\displaystyle W}$ for a given state of the gas, and in this connection there is first required a more precise investigation of that which is to be understood as the state of the gas. Obviously, the state is to be taken here solely in the sense of the conception which we have called macroscopic in the last lecture. Otherwise, a state would possess neither probability nor entropy. Furthermore, we are not allowed to assume a condition of equilibrium for the gas. For this is characterized through the further special condition that the entropy for it is a maximum. Thus, an unequal distribution of density may exist in the gas; also, there may be present an arbitrary number of different currents, and in general no kind of equality between the various velocities of the molecules is to be assumed. The velocities, as the coordinates of the molecules, are rather to be taken a priori as quite arbitrarily given, but in order that the state, considered in a macroscopic sense, may be assumed as known, certain mean values of the densities and the velocities must exist. Through these mean values the state from a macroscopic standpoint is completely characterized.

The conditions mentioned will all be fulfilled if we consider the state as given in such manner that the number of molecules in a sufficiently small macroscopic space, but which, however, contains a very large number of molecules, is given, and furthermore, that the (likewise great) number of these molecules is given, which are found in a certain macroscopically small velocity domain, i. e., whose velocities lie within certain small intervals. If we call the coordinates ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$, and the velocity components ${\displaystyle {\dot {x}}}$, ${\displaystyle {\dot {y}}}$, ${\displaystyle {\dot {z}}}$, then this number will be proportional to[1]

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&dx\cdot dy\cdot dz\cdot d{\dot {x}}\cdot d{\dot {y}}\cdot d{\dot {z}}=\sigma .\end{aligned}}}

It will depend, besides, upon a finite factor of proportionality which may be an arbitrarily given function ${\displaystyle f(x,y,z,{\dot {x}},{\dot {y}},{\dot {z}})}$ of the coordinates and the velocities, and which has only the one condition to fulfill that

{\displaystyle {\begin{aligned}&(14){\color {White}.}\qquad &&{\textstyle \sum }f\cdot \sigma =N,\end{aligned}}}

where ${\displaystyle N}$ denotes the total number of molecules in the gas. We are now concerned with the calculation of the probability ${\displaystyle W}$ of that state of the gas which corresponds to the arbitrarily given distribution function ${\displaystyle f}$.

The probability that a given molecule possesses such coordinates and such velocities that it lies within the domain ${\displaystyle \sigma }$ is expressed, in accordance with the final result of the previous lecture, by the magnitude of the corresponding elementary domain:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&d\varphi _{1}\cdot d\varphi _{2}\cdot d\varphi _{3}\cdot d\psi _{1}\cdot d\psi _{2}\cdot d\psi _{3},\end{aligned}}}

therefore, since here

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&\varphi _{1}=x,\quad \varphi _{2}=y,\quad \varphi _{3}=z,\quad \psi _{1}=m{\dot {x}},\quad \psi _{2}=m{\dot {y}},\quad \psi _{3}=m{\dot {z}},\end{aligned}}}

(${\displaystyle m}$ the mass of a molecule) by

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&m^{3}\sigma .\end{aligned}}}

Now we divide the whole of the six dimensional “state domain” containing all the molecules into suitable equal elementary domains of the magnitude ${\displaystyle m^{3}\sigma }$. Then the probability that a given molecule fall in a given elementary domain is equally great for all such domains. Let ${\displaystyle P}$ denote the number of these equal elementary domains. Next, let us imagine as many dice as there are molecules present, i. e., ${\displaystyle N}$, and each die to be provided with ${\displaystyle P}$ equal sides. Upon these ${\displaystyle P}$ sides we imagine numbers ${\displaystyle 1}$, ${\displaystyle 2}$, ${\displaystyle 3}$, ${\displaystyle \cdots }$ to ${\displaystyle P}$, so that each of the ${\displaystyle P}$ sides indicates a given elementary domain. Then each throw with the ${\displaystyle N}$ dice corresponds to a given state of the gas, while the number of dice which show a given number corresponds to the molecules which lie in the elementary domain considered. In accordance with this, each single die can indicate with the same probability each of the numbers from ${\displaystyle 1}$ to ${\displaystyle P}$, corresponding to the circumstance that each molecule may fall with equal probability in any one of the ${\displaystyle P}$ elementary domains. The probability ${\displaystyle W}$ sought, of the given state of the molecules, corresponds, therefore, to the number of different kinds of throws (complexions) through which is realized the given distribution ${\displaystyle f}$. Let us take, e. g., ${\displaystyle N}$ equal to ${\displaystyle 10}$ molecules (dice) and ${\displaystyle P=6}$ elementary domains (sides) and let us imagine the state so given that there are

 3 molecules in 1st elementary domain 4 molecules in 2d elementary domain 0 molecules in 3d elementary domain 1 molecule in 4th elementary domain 0 molecules in 5th elementary domain 2 molecules in 6th elementary domain,

then this state, e. g., may be realized through a throw for which the 10 dice indicate the following numbers:

 1st 2d 3d 4th 5th 6th 7th 8th 9th 10th 2 6 2 1 1 2 6 2 1 4.

Under each of the characters representing the ten dice stands the number which the die indicates in the throw. In fact,

 3 dice show the figure 1 4 dice show the figure 2 0 dice show the figure 3 1 die shows the figure 4 0 dice show the figure 5 2 dice show the figure 6.

The state in question may likewise be realized through many other complexions of this kind. The number sought of all possible complexions is now found through consideration of the number series indicated in ${\displaystyle (15)}$. For, since the number of molecules (dice) is given, the number series contains a fixed number of elements (${\displaystyle 10=N}$). Furthermore, since the number of molecules falling in an elementary domain is given, each number, in all permissible complexions, appears equally often in the series. Finally, each change of the number configuration conditions a new complexion. The number of possible complexions or the probability ${\displaystyle W}$ of the given state is therefore equal to the number of possible permutations with repetition under the conditions mentioned. In the simple example chosen, in accordance with a well known formula, the probability is

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&{\frac {10!}{3!\;4!\;0!\;1!\;0!\;2!\;}}=12,600.\end{aligned}}}

Therefore, in the general case:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&W={\frac {N!}{\prod (f\cdot \sigma )!}}.\end{aligned}}}

The sign ${\displaystyle \prod }$ denotes the product extended over all of the ${\displaystyle P}$ elementary domains.

From this there results, in accordance with equation ${\displaystyle (13)}$, for the entropy of the gas in the given state:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&S=k\log N!-k{\textstyle \sum }\log(f\cdot \sigma )!.\end{aligned}}}

The summation is to be extended over all domains ${\displaystyle \sigma }$. Since ${\displaystyle f\cdot \sigma }$ is a large quantity, Stirling's formula may be employed for its factorial, which for a large number ${\displaystyle n}$ is expressed by:

{\displaystyle {\begin{aligned}&(16){\color {White}.}\qquad &&n!=\left({\frac {n}{e}}\right)^{n}{\sqrt {2\pi n}},\end{aligned}}}

therefore, neglecting unimportant terms:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&\log n!=n(\log n-1);\end{aligned}}}

and hence:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&S=k\log N!-k{\textstyle \sum }f\sigma (\log[f\cdot \sigma ]-1),\end{aligned}}}

or, if we note that ${\displaystyle \sigma }$ and ${\displaystyle N={\textstyle \sum }f\sigma }$ remain constant in all changes of state:

{\displaystyle {\begin{aligned}&(17){\color {White}.}\qquad &&S={\text{const}}-k{\textstyle \sum }f\cdot \log f\cdot \sigma .\end{aligned}}}

This quantity is, to the universal factor ${\displaystyle (-k)}$, the same as that which L. Boltzmann denoted by ${\displaystyle H}$, and which he showed to vary in one direction only for all changes of state.

In particular, we will now determine the entropy of a gas in a state of equilibrium, and inquire first as to that form of the law of distribution which corresponds to thermodynamic equilibrium. In accordance with the second law of thermodynamics, a state of equilibrium is characterized by the condition that with given values of the total volume ${\displaystyle V}$ and the total energy ${\displaystyle E}$, the entropy ${\displaystyle S}$ assumes its maximum value. If we assume the total volume of the gas

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&V=\int dx\cdot dy\cdot dz,\end{aligned}}}

and the total energy

{\displaystyle {\begin{aligned}&(18){\color {White}.}\qquad &&E={\frac {m}{2}}{\textstyle \sum }({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})f\sigma \end{aligned}}}

as given, then the condition:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&\delta S=0\end{aligned}}}

must hold for the state of equilibrium, or, in accordance with ${\displaystyle (17)}$:

{\displaystyle {\begin{aligned}&(19){\color {White}.}\qquad &&{\textstyle \sum }(\log f+1)\cdot \delta f\cdot \sigma =0,\end{aligned}}}

wherein the variation ${\displaystyle \delta f}$ refers to an arbitrary change in the law of distribution, compatible with the given values of ${\displaystyle N}$, ${\displaystyle V}$ and ${\displaystyle E}$.

Now we have, on account of the constancy of the total number of molecules ${\displaystyle N}$, in accordance with ${\displaystyle (14)}$:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&{\textstyle \sum }\delta f\cdot \sigma =0\end{aligned}}}

and, on account of the constancy of the total energy, in accordance with ${\displaystyle (18)}$:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&{\textstyle \sum }({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})\cdot \delta f\cdot \sigma =0.\end{aligned}}}

Consequently, for the fulfillment of condition ${\displaystyle (19)}$ for all permissible values of ${\displaystyle \delta f}$, it is sufficient and necessary that

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&\log f+\beta ({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})={\text{const}},\end{aligned}}}

or:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&f=\alpha e^{-\beta ({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})},\end{aligned}}}

wherein ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are constants. In the state of equilibrium, therefore, the space distribution of molecules is uniform, i. e., independent of ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$, and the distribution of velocities is the well known Maxwellian distribution.

The values of the constants ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are to be found from those of ${\displaystyle N}$, ${\displaystyle V}$ and ${\displaystyle E}$. For the substitution of the value found for ${\displaystyle f}$ in ${\displaystyle (14)}$ leads to:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&N=V\alpha \left({\frac {\pi }{\beta }}\right)^{\tfrac {3}{2}},\end{aligned}}}

and the substitution of ${\displaystyle f}$ in ${\displaystyle (18)}$ leads to:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&E={\tfrac {3}{4}}Vm{\frac {\alpha }{\beta }}\left({\frac {\pi }{\beta }}\right)^{\tfrac {3}{2}}.\end{aligned}}}

From these equations it follows that:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&\alpha ={\frac {N}{V}}\cdot \left({\frac {3mN}{4\pi E}}\right)^{\tfrac {3}{2}},\quad \beta ={\frac {3mN}{4E}},\end{aligned}}}

and hence finally, in accordance with ${\displaystyle (17)}$, the expression for the entropy ${\displaystyle S}$ of the gas in a state of equilibrium with given values for ${\displaystyle N}$, ${\displaystyle V}$ and ${\displaystyle E}$ is:

{\displaystyle {\begin{aligned}&(20){\color {White}.}\qquad &&S={\text{const}}+kN({\tfrac {3}{2}}\log E+\log V).\end{aligned}}}

The additive constant contains terms in ${\displaystyle N}$ and ${\displaystyle m}$, but not in ${\displaystyle E}$ and ${\displaystyle V}$.

The determination of the entropy here carried out permits now the specification directly of the complete thermodynamic behavior of the gas, viz., of the equation of state, and of the values of the specific heats. From the general thermodynamic definition of entropy:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&dS={\frac {dE+pdV}{T}}\end{aligned}}}

are obtained the partial differential quotients of ${\displaystyle S}$ with regard to ${\displaystyle E}$ and ${\displaystyle V}$ respectively:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&\left({\frac {\partial S}{\partial E}}\right)_{V}={\frac {1}{T}},\quad \left({\frac {\partial S}{\partial V}}\right)_{E}={\frac {p}{T}}.\end{aligned}}}

Consequently, with the aid of ${\displaystyle (20)}$:

{\displaystyle {\begin{aligned}&(21){\color {White}.}\qquad &&\left({\frac {\partial S}{\partial E}}\right)_{V}={\frac {3}{2}}{\frac {kN}{E}}={\frac {1}{T}},\end{aligned}}}

and

{\displaystyle {\begin{aligned}&(22){\color {White}.}\qquad &&\left({\frac {\partial S}{\partial V}}\right)_{E}={\frac {kN}{V}}={\frac {p}{T}}.\end{aligned}}}

The second of these equations:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&p={\frac {kNT}{V}}\end{aligned}}}

contains the laws of Boyle, Gay Lussac and Avogadro, the latter because the pressure depends only upon the number ${\displaystyle N}$, and not upon the constitution of the molecules. Writing it in the ordinary form:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&p={\frac {RnT}{V}},\end{aligned}}}

where ${\displaystyle n}$ denotes the number of gram molecules or mols of the gas, referred to ${\displaystyle O_{2}=32g}$, and ${\displaystyle R}$ the absolute gas constant:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&R=8.315\cdot 10^{7}{\frac {\text{erg}}{\text{deg}}},\end{aligned}}}

we obtain by comparison:

{\displaystyle {\begin{aligned}&(23){\color {White}.}\qquad &&k={\frac {Rn}{N}}.\end{aligned}}}

If we denote the ratio of the mol number to the molecular number by ${\displaystyle \omega }$, or, what is the same thing, the ratio of the molecular mass to the mol mass:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&\omega ={\frac {n}{N}},\end{aligned}}}

and hence:

{\displaystyle {\begin{aligned}&(24){\color {White}.}\qquad &&k=\omega R.\end{aligned}}}

From this, if ${\displaystyle \omega }$ is given, we can calculate the universal constant ${\displaystyle k}$, and conversely.

The equation ${\displaystyle (21)}$ gives:

{\displaystyle {\begin{aligned}&(25){\color {White}.}\qquad &&E={\tfrac {3}{2}}kNT.\end{aligned}}}

Now since the energy of an ideal gas is given by:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&E=Anc_{v}T,\end{aligned}}}

wherein ${\displaystyle c_{v}}$ denotes in calories the heat capacity at constant volume of a mol, ${\displaystyle A}$ the mechanical equivalent of heat:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&A=4.19\cdot 10^{7}{\frac {\text{erg}}{\text{cal}}},\end{aligned}}}

it follows that:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&c_{v}={\frac {3kN}{2An}},\end{aligned}}}

and, having regard to ${\displaystyle (23)}$, we obtain:

{\displaystyle {\begin{aligned}&(26){\color {White}.}\qquad &&c_{v}={\frac {3}{2}}{\frac {R}{A}}=3.0,\end{aligned}}}

the mol heat in calories of any monatomic gas at constant volume.

For the mol heat ${\displaystyle c_{p}}$ at constant pressure we have from the first law of thermodynamics

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&c_{p}-c_{v}={\frac {R}{A}},\end{aligned}}}

and, therefore, having regard to ${\displaystyle (26)}$:

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&c_{p}=5,\quad {\frac {c_{p}}{c_{v}}}={\tfrac {5}{3}},\end{aligned}}}

a known result for monatomic gases.

The mean kinetic energy ${\displaystyle L}$ of a molecule is obtained from ${\displaystyle (25)}$:

{\displaystyle {\begin{aligned}&(27){\color {White}.}\qquad &&L={\frac {E}{N}}={\tfrac {3}{2}}kT.\end{aligned}}}

You notice that we have derived all these relations through the identification of the mechanical with the thermodynamic expression for the entropy, and from this you recognize the fruitfulness of the method here proposed.

But a method can first demonstrate fully its usefulness when we utilize it, not only to derive laws which are already known, but when we apply it in domains for whose investigation there at present exist no other methods. In this connection its application affords various possibilities. Take the case of a monatomic gas which is not sufficiently attenuated to have the properties of the ideal state; there are here, as pointed out by J. D. van der Waals, two things to consider: (1) the finite size of the atoms, (2) the forces which act among the atoms. Taking account of these involves a change in the value of the probability and in the energy of the gas as well, and, so far as can now be shown, the corresponding change in the conditions for thermodynamic equilibrium leads to an equation of state which agrees with that of van der Waals. Certainly there is here a rich field for further investigations, of greater promise when experimental tests of the equation of state exist in larger number.

Another important application of the theory has to do with heat radiation, with which we shall be occupied the coming week. We shall proceed then in a similar way as here, and shall be able from the expression for the entropy of radiation to derive the thermodynamic properties of radiant heat.

Today we will refer briefly to the treatment of polyatomic gases. I have previously, upon good grounds, limited the treatment to monatomic molecules; for up to the present real difficulties appear to stand in the way of a generalization, from the principles employed by us, to include polyatomic molecules; in fact, if we wish to be quite frank, we must say that a satisfactory mechanical theory of polyatomic gases has not yet been found. Consequently, at present we do not know to what place in the system of theoretical physics to assign the processes within a molecule—the intra-molecular processes. We are obviously confronted by puzzling problems. A noteworthy and much discussed beginning was, it is true, made by Boltzmann, who introduced the most plausible assumption that for intra-molecular processes simple laws of the same kind hold as for the motion of the molecules themselves, i. e., the general equations of dynamics. It is easy then, in fact, to proceed to the proof that for a monatomic gas the molecular heat ${\displaystyle c_{v}}$ must be greater than ${\displaystyle 3}$ and that consequently, since the difference ${\displaystyle c_{p}-c_{v}}$ is always equal to ${\displaystyle 2}$, the ratio is

{\displaystyle {\begin{aligned}&{\color {White}.(00)}\qquad &&{\frac {c_{p}}{c_{v}}}={\frac {c_{v}+2}{c_{v}}}<{\tfrac {5}{3}}.\end{aligned}}}

This conclusion is completely confirmed by experience. But this in itself does not confirm the assumption of Boltzmann; for, indeed, the same conclusion is reached very simply from the assumption that there exists intra-molecular energy which increases with the temperature. For then the molecular heat of a polyatomic gas must be greater by a corresponding amount than that of a monatomic gas.

Nevertheless, up to this point the Boltzmann theory never leads to contradiction with experience. But so soon as one seeks to draw special conclusions concerning the magnitude of the specific heats hazardous difficulties arise; I will refer to only one of them. If one assumes the Hamiltonian equations of mechanics as applicable to intra-molecular motions, he arrives of necessity at the law of “uniform distribution of energy,” which asserts that under certain conditions, not essential to consider here, in a thermodynamic state of equilibrium the total energy of the gas is distributed uniformly among all the individual energy phases corresponding to the independent variables of state, or, as one may briefly say; the same amount of energy is associated with every independent variable of state. Accordingly, the mean energy of motion of the molecules ${\displaystyle {\tfrac {1}{2}}kT}$, corresponding to a given direction in space, is the same as for any other direction, and, moreover, the same for all the different kinds of molecules, and ions; also for all suspended particles (dust) in the gas, of whatever size, and, furthermore, the same for all kinds of motions of the constituents of a molecule relative to its centroid. If one now reflects that a molecule commonly contains, so far as we know, quite a large number of different freely moving constituents, certainly, that a normal molecule of a monatomic gas, e. g., mercury, possesses numerous freely moving electrons, then, in accordance with the law of uniform energy distribution, the intra-molecular energy must constitute a much larger fraction of the whole specific heat of the gas, and therefore ${\displaystyle c_{p}/c_{v}}$ must turn out much smaller, than is consistent with the measured values. Thus, e. g., for an atom of mercury, in accordance with the measured value of ${\displaystyle c_{p}/c_{v}=5/3}$, no part whatever of the heat added may be assigned to the intra-molecular energy. Boltzmann and others, in order to eliminate this contradiction, have fixed upon the possibility that, within the time of observation of the specific heats, the vibrations of the constituents (of a molecule) do not change appreciably with respect to one another, and come later with their progressive motion so slowly into heat equilibrium that this process is no longer capable of detection through observation. Up to now no such delay in the establishment of a state of equilibrium has been observed. Perhaps it would be productive of results if in delicate measurements special attention were paid the question as to whether observations which take a longer time lead to a greater value of the mol-heat, or, what comes to the same thing, a smaller value of ${\displaystyle c_{p}/c_{v}}$, than observations lasting a shorter time.

If one has been made mistrustful through these considerations concerning the applicability of the law of uniform energy distribution to intra-molecular processes, the mistrust is accentuated upon the inclusion of the laws of heat radiation. I shall make mention of this in a later lecture.

When we pass from stable atoms to the unstable atoms of radioactive substances, the principles following from the kinetic gas theory lose their validity completely. For the striking failure of all attempts to find any influence of temperature upon radioactive phenomena shows us that an application here of the law of uniform energy distribution is certainly not warranted. It will, therefore, be safest meanwhile to offer no definite conjectures with regard to the nature and the laws of these noteworthy phenomena, and to leave this field for further development to experimental research alone, which, I may say, with every day throws new light upon the subject.

1. We can call ${\displaystyle \sigma }$ a “macro-differential” in contradistinction to the micro-differentials which are infinitely small with reference to the dimensions of a molecule. I prefer this terminology for the discrimination between “physical” and “mathematical” differentials in spite of the inelegance of phrasing, because the macro-differential is also just as much mathematical as physical and the micro-differential just as much physical as mathematical.


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#### A Quasi-PTAS for Profit-Maximizing Pricing on Line Graphs

##### MPS-Authors
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44374

Elbassioni,  Khaled
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45495

Sitters,  Rene
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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##### Citation

Elbassioni, K., Sitters, R., & Zhang, Y. (2007). A Quasi-PTAS for Profit-Maximizing Pricing on Line Graphs. In L. Arge, M. Hoffmann, & E. Welzl (Eds.), Algorithms - ESA 2007: 15th Annual European Symposium (pp. 451-462). Berlin, Germany: Springer.

Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-1E25-B
##### Abstract
We consider the problem of pricing items so as to maximize the profit made from selling these items. An instance is given by a set $E$ of $n$ items and a set of $m$ clients, where each client is specified by one subset of $E$ (the bundle of items he/she wants to buy), and a budget (valuation), which is the maximum price he is willing to pay for that subset. We restrict our attention to the model where the subsets can be arranged such that they form intervals of a line graph. Assuming an unlimited supply of any item, this problem is known as \emph{the highway problem} and so far only an $O(\log n)$-approximation algorithm is known. We show that a PTAS is likely to exist by presenting a quasi-polynomial time approximation scheme. We also combine our ideas with a recently developed quasi-PTAS for the unsplittable flow problem on line graphs to extend this approximation scheme to the limited supply version of the pricing problem.

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# Find the limit of the expression $\lim_{x\to 0}\left(\frac{\sin x}{\arcsin x}\right)^{1/\ln(1+x^2)}$

Limit: $$\lim_{x\to 0}\left(\dfrac{\sin x}{\arcsin x}\right)^{1/\ln(1+x^2)}$$ I have tried to do this: it is equal to $$e^{\lim\frac{\log{\frac{\sin x}{\arcsin x}}}{\log(1+x^2)}}$$, but I can't calculate this with the help of l'Hopital rule or using Taylor series, because there is very complex and big derivatives, so I wish to find more easier way. $$\lim_{x\rightarrow 0}{\frac{\log{\frac{\sin x}{\arcsin x}}}{\log(1+x^2)}} = \lim_{x\rightarrow 0}\frac{\log1 + \frac{-\frac{1}{3}x^2}{1+\frac{1}{6}x^2+o(x^2)}}{\log(1+x^2)} = \lim_{x\rightarrow0}\frac{\frac{-\frac{1}{3}x^2}{1+\frac{1}{6}x^2+o(x^2)} + o(\frac{-\frac{1}{3}x^2}{1+\frac{1}{6}x^2+o(x^2)})}{x^2+o(x^2)}$$ using Taylor series. Now I think that it's not clear for me how to simplify $$o\left(\frac{-\frac{1}{3}x^2}{1+\frac{1}{6}x^2+o(x^2)}\right)$$.

• This is not a homework solving site. Please see the help center. In particular, show effort and expect to do the work – Brevan Ellefsen Nov 29 '18 at 7:57
• Why don't you try the usual approach of taking logarithm and let us know if you face any problems? Just giving a problem statement is not encouraged here. – Paramanand Singh Nov 29 '18 at 7:59
• Now that you have effectively taken logs and also used $e$ just focus on the exponent itself. Do you recall any standard limits by looking at the denominator $\log(1+x^2)$? – Paramanand Singh Nov 29 '18 at 8:10
• If you look carefully both numerator and denominator are log expressions which tend to $0$ and hence they can be simplified by the use of the same standard limit and you should try to proceed in that manner. – Paramanand Singh Nov 29 '18 at 8:16
• @J_G that first step seems wrong$$\lim_{x\rightarrow 0}{\frac{\log{\frac{\sin x}{\arcsin x}}}{\log(1+x^2)}} = \lim_{x\rightarrow 0}\frac{\log1 + \frac{-\frac{1}{3}x^2}{1+\frac{1}{6}x^2+o(x^2)}}{\log(1+x^2)}...$$ – user Nov 29 '18 at 8:30

HINT

By Taylor's series

$$\frac{\sin x}{\arcsin x}=\frac{x-\frac16x^3+o(x^3)}{x+\frac16x^3+o(x^3)}=\frac{1-\frac16x^2+o(x^2)}{1+\frac16x^2+o(x^2)}=$$$$=\left(1-\frac16x^2+o(x^2)\right)\left(1+\frac16x^2+o(x^2)\right)^{-1}$$

Can you continue form here using binomial series for the last term?

• This won't help as it leads to indeterminate form $1^{\infty}$. Or may be you wanted to emphasize that it is a particular indeterminate form. – Paramanand Singh Nov 29 '18 at 8:01
• @ParamanandSingh Yes it was just a hint to start with. Do you think it too small? – user Nov 29 '18 at 8:05
• It will help if the asker knows how to deal with $1^{\infty}$ and unless the asker says anything we can't be sure. – Paramanand Singh Nov 29 '18 at 8:07
• @ParamanandSingh After the editing, I've added a different hint to start with using Taylor's series. – user Nov 29 '18 at 8:12

So you want to find the limit $$L=\lim\limits_{x\to0} \frac{\ln\frac{\sin x}{\arcsin x}}{\ln(1+x^2)}.$$ Perhaps a reasonable strategy would be to split this into calculating several simpler limits. We know that $$\begin{gather*} \lim\limits_{x\to0} \frac{\ln\frac{\sin x}{\arcsin x}}{\frac{\sin x}{\arcsin x}-1}=1\\ \lim\limits_{x\to0} \frac{\ln(1+x^2)}{x^2}=1 \end{gather*}$$ so we eventually get to the limit $$L=\lim\limits_{x\to0} \frac{\frac{\sin x}{\arcsin x}-1}{x^2} =\lim\limits_{x\to0} \frac{\sin x-\arcsin x}{x^2\arcsin x}.$$ If we also use that $$\lim\limits_{x\to0} \frac{\arcsin x}x=1$$, we get that $$L=\lim\limits_{x\to0} \frac{\sin x-\arcsin x}{x^3}.$$ And now we can try to calculate separately the two limits \begin{align*} L_1&=\lim\limits_{x\to0} \frac{\sin x-x}{x^3}\\ L_2&=\lim\limits_{x\to0} \frac{x-\arcsin x}{x^3} \end{align*} Both $$L_1$$ and $$L_2$$ seem as limits where L'Hospital's rule or Taylor expansion should lead to result. In fact, substitution $$y=\sin x$$ transforms $$L_2$$ to a limit very similar to $$L_1$$.

You can probably find also some posts on this site at least for $$L_1$$ (and maybe also for $$L_2$$). For example: Solve $$\lim_{x\to 0} \frac{\sin x-x}{x^3}$$, Find the limit $$\lim_{x\to0}\frac{\arcsin x -x}{x^2}$$, Are all limits solvable without L'Hôpital Rule or Series Expansion.

• +1 this is what I was suggesting in comments. A little algebraic manipulation combined with standard limits always helps. – Paramanand Singh Nov 30 '18 at 2:45


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# Mathematical programs with complementarity constraints: convergence properties of a smoothing method

G. Bouza Allende, Georg J. Still

Research output: Contribution to journalArticleAcademicpeer-review

15 Citations (Scopus)

## Abstract

In this paper, optimization problems $P$ with complementarity constraints are considered. Characterizations for local minimizers $\bar{x}$ of $P$ of Orders 1 and 2 are presented. We analyze a parametric smoothing approach for solving these programs in which $P$ is replaced by a perturbed problem $P_{\tau}$ depending on a (small) parameter $\tau$. We are interested in the convergence behavior of the feasible set $\cal{F}_{\tau}$ and the convergence of the solutions $\bar{x}_{\tau}$ of $P_{\tau}$ for $\tau\to 0.$ In particular, it is shown that, under generic assumptions, the solutions $\bar{x}_{\tau}$ are unique and converge to a solution $\bar{x}$ of $P$ with a rate $\cal{O}(\sqrt{\tau})$. Moreover, the convergence for the Hausdorff distance $d(\cal{F}_{\tau}$, $\cal{F})$ between the feasible sets of $P_{\tau}$ and $P$ is of order $\cal{O}(\sqrt{\tau})$.
Original language Undefined 10.1287/moor.1060.0245 467-483 17 Mathematics of operations research 32 LNCS4549/2 https://doi.org/10.1287/moor.1060.0245 Published - May 2007

• EWI-11089
• MSC-65K05
• IR-64357
• MSC-90C33
• METIS-241917
• MSC-90C30


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# ExpressionSpaceFunctions2(E, F)¶

This package allows a mapping E -> F to be lifted to a kernel over E; This lifting can fail if the operator of the kernel cannot be applied in F; Do not use this package with E = F, since this may drop some properties of the operators.
map(f, k) returns g = op(f(a1), ..., f(an)) where k = op(a1, ..., an).


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Practical LaTeX

Since Donald Knuth gave the scientific community TeX and Leslie Lamport designed LaTeX on top, publishing and typesetting mathematics was revolutionized. Many packages and variations like AMS-LaTeX were designed on top of these. Nowadays many math journals provide LaTeX style files for typesetting your manuscript. The advantage is that the underlying TeX language is very stable, and the properly edited source file is readable and is very structured. This is very much unlike point-and-click WYSIWYG type of word processing systems. Numerous are the books available, documenting the extra packages and explaining the use of LaTeX, both on paper and on the Web (see e.g. the online wikibook). On different discussion forums, one may find many different opinions on which is the best book to learn LaTeX from. Which one best suites you will probably depend on your background and previous experience with typing and rendering manuscripts, but also on how urgent and/or how frequent you will need to use LaTeX, and/or what kind of exotic typesetting you need. In my opinion, for conventional mathematical papers, there is nothing surpassing LaTeX in one form or another. The problem with many of these books is often that they want to be as complete as possible and they describe all the features of the whole LaTeX language. This gives the impression of a quite steep learning curve, which it need not be. Grätzer, the author of this book, is also the author of one of these more "extensive" documents More Math into LaTeX (Springer, 2007). With its 620 pages, it is too much to get started.

Like so many other authors, Grätzer also wants to provide a shorter start-up introduction. Disregarding the appendices and the (extensive) index, the proper material of the extensive version has been reduced from 488 to 160 pages in this starters version. It is basically a revision of his earlier First Steps in LaTeX book (Birkhäuser, 1999). The book starts with a quick survey, and then explores a bit deeper how to typeset the text, the use of environments, (mathematical) formulas and arrays, and finally the global structure of the document (top matter, body, back matter). That should have you going a long way. Only then the reader is informed about defining his or her own commands and environments. For presentations, there is a short introduction to a "baby-version" of beamer, and a minimal description of TikZ to generate plots (or e.g. commuting diagrams). Grätzer has chosen for an AMS-like description and takes the amsart class as the basic one for typesetting the manuscript. Essential are the tables with text and math symbol commands that are listed as appendices. Certainly as a beginner, you will have to lookup these tables a lot. Somewhat surprising is the third appendix in which it is explained how you may get LaTeX running on your iPad. I am not as convinced as Grätzer seems to be that typesetting LaTeX documents on your iPad is a viable alternative. In exceptional emergency situations it might be an option, but still I doubt it.

Of course, reducing the full version to less than half, requires some shortcuts. For example downloading and installing the MikTex system (a LaTeX distribution) is almost disregarded, and the help of a colleague or friend is advised if it is too difficult for you. There are of course many alternatives to generate plots, the BibTeX description for bibliographies is very short, and for MakeIndex to generate an index one is referred to the extensive book. What is however quite interesting is that he manages to include his experience as a user into this booklet by inserting many "practical tips". These are guidelines to learn good practice from the start or to prevent errors that beginners will easily make, how to locate an error, or what may have caused a particular error, etc. Of course many sample tex files are available via the website of the book.

This may be seen as an alternative for several other introductions to LaTeX. For example the The Not So Short Introduction to LaTeX2ε by Tobias Oetiker that is available on the documentation site of the LaTeX project. That is more focused on the plain LaTeX2ε, while here focus is more on the AMS-LaTeX. Also the journals in which you want to publish that have their own style files will often provide a general introduction to use their package, although there is a basic package like AMS-(La)TeX underneath. Thus when it comes to the proper content of your paper, you will need more, and then, if you are a novice, this can be provided by a booklet like this one. Although, I have been using LaTeX for many years to produce many papers and even books, and I consider myself relatively experienced, there is always something to learn, even from an introduction like this one. After some time one is frozen in those ingrained habits of doing things, and it needs another vision to find out about alternative or better ways to do it. Hence this book might be interesting to read, not only for the beginner, but also for the experienced LaTeX user.

Reviewer:
Book details

This is a practical "beginners" guide to start using LaTeX to produce papers and presentations.

Publisher:

Published:
2014
ISBN:
978-3-319-06424-6 (pbk)
Price:
28,61 € (pbk)
Pages:
216
Categorisation


text: Center for                           Nonlinear Analysis CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium Chanwoo Kim University of Wisconsin Title: Stationary solutions to the Boltzmann equation in the Hydrodynamic limit Abstract: Despite its conceptual and practical importance, the rigorous derivation of the steady incompressible Navier-Stokes-Fourier system from the Boltzmann theory has been an outstanding open problem for general domains in 3D. We settle this open question in the affirmative, in the presence of a small external field and a small boundary temperature variation for the diffuse boundary condition. Avoiding the boundary layer correction to cope with general geometry, we employ a recent quantitative $L^2-L^\infty$ approach with crucial $L^3$ the hydrodynamic part of the distribution function. Such a gain of integrability is established via an extension lemma for the non-grazing component of the distribution function, which is critical to close our program in 3D. Our results also imply the validity of Fourier law in the hydrodynamical limit, and our method leads to asymptotical stability of steady Boltzmann solutions as well as the derivation of the unsteady Navier-Stokes-Fourier system.Recording: http://mm.math.cmu.edu/recordings/cna/CNA-ChanwooKim-Mar-31-2015.aviRecording: http://mm.math.cmu.edu/recordings/cna/CNA-ChanwooKim-Mar-31-2015.mp4Date: Tuesday, March 31, 2015Time: 1:30 pmLocation: Wean Hall 7218Submitted by:  David Kinderlehrer


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## anonymous 5 years ago How do you find the differential of the integral cos^3(x) dx from -9ts to -5t+6s?

1. anonymous

$\int\limits_{-9ts}^{-5t+6s}\cos ^{3}(x) dx$

2. amistre64

ummmm......... i dont really know...

3. amistre64

i spose its possible, just use those"equations" as the F(x) stuff in the end

4. amistre64

you might want to go about "reducing" the integrand to an addition of sin and cos....maybe

5. amistre64

cos cos^2 cos(1-sin^2) cos(x)(1 - (1-cos(2x)/2)) ... along those lines eh?

6. anonymous

yeah thats what i was going to say

7. amistre64

well say it again then :)

8. anonymous

integral of sin^2 is known as -x/2 + sincos/2 so you could use that

9. anonymous

Well I have notes and an example problem, but it doesnt really make sense. He has the differential of the integral e^-x^2 from a to b. He then takes partials. Fa = -e^a^2 and Fb = e^b^2. Then apparently the differential is -e^a^2 * da + e^b^2 * db. No sense.

10. anonymous

Don't worry about it, I just wondered if anyone could make sense out of it, but obviously my teacher is drunk. He wrote his own "textbook" in Mathematica and he didn't do a very good job.

11. anonymous

are you still talking about integral of cos^3? anyway yeah there are no elementary functions to describe integral of e^(-x^2).


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# zbMATH — the first resource for mathematics

The strong law of large numbers for weighted averages under dependence assumptions. (English) Zbl 0857.60021
The authors prove strong laws of large numbers for weighted averages of dependent random variables, generalizing the classical work of B. Jamison, S. Orey and W. Pruitt [Z. Wahrscheinlichkeitstheorie Verw. Geb. 4, 40-44 (1965; Zbl 0141.16404)] for i.i.d. sequences. The dependence structure imposed is asymptotic quadrant sub-independent, requiring that $P(X_i>s, X_j>t) - P(X_i>s) P(X_j>t) \leq q \bigl(|i-j |\bigr) \alpha_{ij} (s,t),$ together with a similar condition on $$P(X_i<s, X_j<t)$$. This condition generalizes the notion of asymptotic quadrant independence, introduced by T. Birkel [Stat. Probab. Lett. 7, No. 1, 17-20 (1988; Zbl 0661.60048)]. The authors also prove a Marcinkiewicz-Zygmund SLLN for weighted averages. The proofs make heavy use of unpublished results by the same authors.

##### MSC:
 60F15 Strong limit theorems
##### Keywords:
law of large numbers; summability methods; weak dependence
Full Text:
##### References:
 [1] Birkel, T. (1992). Laws of large numbers under dependence assumptions.Statist. Prob. Lett. 14, 355–362. · Zbl 0925.60023 [2] Chandra, T. K. (1991). Extensions of Rajchman’s strong law of large numbers.Sankhyā, Ser. A 53, 118–121. · Zbl 0749.60028 [3] Chandra, T. K., and Ghosal, S. (1993). Some elementary strong laws of large numbers: a review. Technical Report #22/93, Indian Statistical Institute. · Zbl 0928.60020 [4] Chandra, T. K., and Ghosal, S. (1996). Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund.Acta Math. Hung. 72(3) (to appear). · Zbl 0853.60032 [5] Etemadi, N. (1983). Stability of sums of weighted random variables.J. Multivariate Anal. 13, 361–365. · Zbl 0531.60034 [6] Hall, P., and Heyde, C. C. (1980).Martingale Limit Theory and Its Application., Academic Press, New York. · Zbl 0462.60045 [7] Jamison, B., Orey, S., and Pruitt, W. E. (1965). Convergence of weighted averages of independent random variables.Z. Wahrsch. Verw. Gebiete 4, 40–44. · Zbl 0141.16404 [8] McLeish, D. L. (1975). A maximal inequality and dependent strong laws.Ann. Prob. 3, 829–839. · Zbl 0353.60035 [9] Pruitt, W. E. (1966). Summability of independent random variables.J. Math. Mech. 15, 769–776. · Zbl 0158.36403 [10] Rothatgi, V. K. (1971). Convergence of weighted sums of independent random variables.Proc. Cambridge Phil. Soc. 69, 305–307. · Zbl 0209.20004 [11] Rosalsky, A. (1987). Strong stability of normed sums of pairwise i.i.d. random variables.Bull. Inst. Math. Acad. Sinica 15, 203–219. · Zbl 0629.60034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# Decrypting a Message Encrypted in RSA Using Two Coprime Encryption Keys

The last question of our number theory final review is as follows:

The same plaintext $P$ is encrypted in RSA using two coprime encryption keys $e_1$, $e_2$. Show how this message can be decrypted quickly if both ciphertexts $C_1$, $C_2$ are intercepted.

Hint: Bézout's identity.

I know the RSA cryptosystem has it that

\begin{align} C_1&\equiv P^{e_1}\pmod{n_1}\text{ and}\\ C_2&\equiv P^{e_2}\pmod{n_2}, \end{align}

where $n_1=p_1q_1$, $n_2=p_2q_2$, $(e_1,\varphi(n_1))=1$, and $(e_2,\varphi(n_2))=1$, for some large primes $p_1$, $q_1$, $p_2$, and $q_2$.

Since one does not have access to the secret keys $d_1$ and $d_2$, I assume Bézout's identity must play a big role here. However, the only place I can think of using it is with the encryption keys

\begin{align} 1&=e_1s_1+n_1t_1,\\ 1&=e_2s_2+n_2t_2,\text{ and}\\ 1&=e_1s_3+e_2t_3, \end{align}

for some integers $s_1$, $t_1$, $s_2$, $t_2$, $s_3$, and $t_3$.

Nevertheless, this is not very useful. What am I missing?

-
Use the third one, relating $e_1$ and $e_2$. –  Ted May 6 '12 at 2:10
I suppose then that one can find the plaintext $P$ by multiplying $C_1$ by $(e_2t_3)/s_3$? Is it that simple? –  Josué Molina May 6 '12 at 2:33
Are you sure that two different moduli $n_1$ and $n_2$ are used? Perhaps you just have a single $n=pq$ used in both encryptions, so that $C_1 \equiv P^{e_1} \pmod{n}$ and $C_2 \equiv P^{e_2} \pmod{n}$. –  Jonas Kibelbek May 6 '12 at 2:47
@JonasKibelbek, the question doesn't specify that, but you could be right, and forgive my ignorance, but what difference would different moduli make? –  Josué Molina May 6 '12 at 2:53
@JosuéMolina: The solution actually does not require taking roots modulo $n$. When you write $C_2^{t/s}$, it looks like you are asking for an $s$-th root modulo $n$. This is a problem, because if $(s,\varphi(n))\neq 1$, then not every residue class has a unique $s$-th root. Every residue class does have a unique $e_2$-th root, but finding it is very difficult in general-- this is the strength of RSA encryption. –  Jonas Kibelbek May 6 '12 at 3:27
show 1 more comment

I suspect that the messages $C_1$ and $C_2$ were encrypted with the same modulus $n=pq$, even though different exponents $e_1$ and $e_2$ were used. Then we have $$C_1 \equiv P^{e_1} \pmod{n}$$ $$C_2 \equiv P^{e_2} \pmod{n}.$$ Further, we can find integers $s$ and $t$ such that $se_1+te_2 =1$.
Can you figure out how to obtain $P^{se_1+te_2} \pmod{n}$? Can you see why this idea works if the same modulus $n$ is used for both encryptions, but does not work if we have two different moduli $n_1$ and $n_2$?
I ended up with $P\equiv C_1^s\cdot C_2^t$. –  Josué Molina May 6 '12 at 3:24
Yes, exactly. One of the integers $s$ and $t$ will be negative. So to calculate $P$, you will need to find an inverse of $C_1$ or of $C_2$ modulo $n$, do modular exponentiation, and multiply two numbers modulo $n$. All of these steps are very fast, so you have found an easy way to calculate $P$. –  Jonas Kibelbek May 6 '12 at 3:31


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## Wednesday, April 16, 2014

### Another anti-physics issue of SciAm

High energy physics is undoubtedly the queen and the ultimate reductionist root of all natural sciences. Nevertheless, during the last decade, it has become immensely fashionable for many people to boast that they're physics haters.

The cover of the upcoming May 2014 issue of Scientific American looks doubly scary for every physicist who has been harassed by the communist regime. It resembles a Soviet flag with some deeply misleading propaganda written over it:
A crisis in physics?

If supersymmetry doesn't pan out, scientists need a new way to explain the universe. [In between the lines]
Every part of this claim is pure bullshit, of course. First of all, there is no "crisis in physics". Second of all, chances are high that we won't be any certain whether SUSY is realized in Nature. Either SUSY will be found at the LHC in 2015 or soon afterwards, or it won't be. In the latter case, the status of SUSY will remain qualitatively the same as it is now. Top-down theorists will continue to be pretty much certain that SUSY exists in Nature in one form or another, one scale or another; bottom-up phenomenologists and experimenters will increasingly notice the absence of evidence – which is something else than the evidence for absence, however.

But aside from this delusion, the second part of the second sentence is totally misguided, too. Supersymmetry isn't a "new way to explain the universe". It is another symmetry, one that differs from some other well-known symmetries such as the rotational or Lorentz symmetry by its having fermionic generators but one that doesn't differ when it comes to its being just one aspect of theories. Supersymmetry isn't a theory of the universe by itself (in the same sense as the Standard Model or string theory); supersymmetry is a feature of some candidate theories of the universe.

To be sure that the hostility is repetitive (a lie repeated 100 times becomes the truth, she learned from Mr Goebbels), editor-in-chief Ms Mariette DiChristina introduces the May 2014 issue under the following title:
Does Physics Have a Problem?
What does it even mean for a scientific discipline to have a problem? Claims in science are either right or wrong. Some theories turn out to be right (at least temporarily), others turn out to be wrong. Some theories are viable and compatible with the evidence, others have been falsified. Some scientists are authors of right and/or important and valuable theories, others are authors of wrong ones or no theories at all.

Some classes of questions are considered settled so they are not being researched as "hot topics" anymore; others are behind the frontier where the scientists don't know the answers (and sometimes the questions): they are increasingly confused by the questions behind the frontier. This separation of the realm of questions by a fuzzy frontier of ignorance is a feature of science that applies to every scientific discipline and every moment of its history. One could argue that there can't be "crises in physics" at all but it's doubly bizarre to use this weird word for the current era which is as ordinary era of normal science as one can get.

The main article about popular physics was written by experimenter Maria Spiropulu (CMS, Caltech) and phenomenologist Joseph Lykken (a self-described very smart guy at Fermilab). They're very interesting and sensible folks but I would have objections to many things they wrote down and I think that the same thing holds for most high energy physicists.

They say that most HEP physicists believe that SUSY is true but add:
Indeed, results from the first run of the LHC have ruled out almost all the best-studied versions of supersymmetry. The negative results are beginning to produce if not a full-blown crisis in particle physics, then at least a widespread panic. The LHC will be starting its next run in early 2015, at the highest energies it was designed for, allowing researchers at the ATLAS and CMS experiments to uncover (or rule out) even more massive superpartners. If at the end of that run nothing new shows up, fundamental physics will face a crossroads: either abandon the work of a generation for want of evidence that nature plays by our rules, or press on and hope that an even larger collider will someday, somewhere, find evidence that we were right all along…
I don't have – and I have never had – any strong preference concerning the masses of superpartners i.e. the accessibility of SUSY by the collider experiments. All of them could have been below $$100\GeV$$ but they may be at $$100\TeV$$ or near the GUT scale, too. Naturalness suggests that they (especially the top squarks, higgsinos, and perhaps gluinos) are closer to the Higgs mass but it is just a vague argument based on Bayesian reasoning that is moreover tied to some specific enough models. Any modification of the SUSY model changes the quantification of the fine-tuning.

But even if it doesn't, the word "natural" is a flexible adjective. If the amount of fine-tuning increases, the model doesn't become unnatural instantly. It is a gradual change. What I find preposterous is the idea presented by the authors that "if the 2015 LHC run finds no proof of SUSY, fundamental physics will face a crossroads; it will either abandon the work altogether or press for a bigger collider".

You can make a 2016 New Year's resolution and say that you will stop thinking about SUSY if there is no evidence from the LHC for SUSY by that time. You may even establish a sect within high energy physics that will share this New Year's resolution with you. But it is just a New Year's resolution, not a science or a decision "implied" by the evidence. There will be other people who will consider your group's New Year's resolution to be premature and just downright stupid. Physics isn't organized by deadlines or five-year plans.

Other people will keep on working on some SUSY models because these models will be attractive and compatible with all the evidence available at that moment. Even if SUSY were experimentally proven to require a 1-in-1,000 fine-tuning – and it really can't be due to the model-dependence of the fine-tuning scores – most people will still rationally think that a 1-in-1,000 fine-tuning is better than the 1-in-1,000,000,000,000,000,000,000,000,000,000 fine-tuning apparently required by the Standard Model. Maria and Joseph know that it is so. In fact, they explicitly mention the "prepared reaction" by Nima Arkani-Hamed that Nima presented in Santa Barbara:
What if supersymmetry is not found at the LHC, [Nima] asked, before answering his own question: then we will make new supersymmetry models that put the superpartners just beyond the reach of the experiments. But wouldn’t that mean that we would be changing our story? That’s okay; theorists don’t need to be consistent—only their theories do.
If SUSY looks attractive enough, of course that phenomenologists will ignore the previous fashionable beliefs about the lightness of the superpartners and (invent and) focus on new models that are compatible with all the evidence at that moment. The relative fraction of hep-ph papers that are dedicated to SUSY model building may decrease in the case of the continuing absence of evidence but only gradually so simply because there are no any major enough alternatives that could completely squeeze the SUSY research. There can't really be any paradigm shift if the status quo continues. You either need some new experimental discoveries or some new theoretical discoveries for a paradigm shift.
This unshakable fidelity to supersymmetry is widely shared. Particle theorists do admit, however, that the idea of natural supersymmetry is already in trouble and is headed for the dustbin of history unless superpartners are discovered soon…
The word "natural" has several meanings and the important differences between these meanings is being (deliberately?) obfuscated by this sentence. It is almost a tautology that any theory that ultimately describes Nature accurately is "natural". But as long as we are ignorant about all the details about the final theory and how it describes Nature, we must be satisfied with approximate and potentially treacherous but operationally applicable definitions of "naturalness". In effective field theory, we assume that the parameters (at the high energy scale) are more or less uniformly distributed in a set and classify very special, unlikely (by this probability distribution) regions to be "unnatural" (typically very small values of some dimensionless parameters that could be of order one).

But the ultimate theory has different rules how to calculate the "probability distribution for the parameters". After all, string theory implies discrete values of all the parameters, so with some discrete information, we may sharpen the probability distribution for low-energy parameters to a higher-dimensional delta-function. We can just calculate the values of all the parameters. The values may be generic or natural according to some sensible enough smooth probability distribution (e.g. in an effective field theory). But if the effective field theory description overlooks some important new particles, interactions, patterns, or symmetries, it may be unnatural, too.

It's important to realize that our ways to estimate whether some values of parameters in some theories are natural are model-dependent and therefore bound to evolve. It is just completely wrong for Maria and Joseph to impose some ideas about physics from some year – 2000 or whatever is the "paradigm" they want everyone to be stuck at – and ban any progress of the thinking. Scientists' thinking inevitably evolves. That's why the scientific research is being done in the first place. So new evidence – including null results – is constantly being taken into account as physicists are adjusting their subjective probabilities of various theories and models, and of various values of parameters within these models.

This process will undoubtedly continue in 2015 and 2016 and later, too. At least, sensible people will continue to adjust their beliefs. If you allow me to say a similar thing as Nima did: theorists are not only allowed to present theories that are incompatible with some of their previous theories or beliefs. They are really obliged to adjust their beliefs – and even at one moment, a sensible enough theorists may (and perhaps should) really be thinking about many possible theories, models, and paradigms. Someone whose expectations turn out to be more accurate and nontrivially agreeing with the later observations should become more famous than others. But it is not a shame to update the probabilities of theories according to the new evidence. It's one of the basic duties that a scientist has to do!

I also feel that the article hasn't taken the BICEP2 results into account and for those reasons, it will already be heavily obsolete when the issue of Scientific American is out. They try to interpret the null results from the LHC as an argument against grand unification or similar physics at the GUT scale. But nothing like that follows from the null results at the LHC and in fact, the BICEP2's primordial gravitational waves bring us quite powerful evidence – if not a proof – that new interesting physics is taking place near the usual GUT scale i.e. not so far from the standard four-dimensional Planck scale.

So in the absence of the SM-violating collider data, the status quo will pretty much continue and the only other way to change it is to propose some so far overlooked alternative paradigm to SUSY that will clarify similar puzzles – or at least a comparable number of puzzles – as SUSY. It is totally plausible that bottom-up particle model builders will have to work with the absence of new collider discoveries – top-down theorists have worked without them for decades, anyway. It works and one can find – and string theorists have found – groundbreaking things in this way, too.

What I really dislike about the article is that – much like articles by many non-physicists – it tries to irrationally single out SUSY as a scapegoat. Even if one should panic about the null results from the LHC, and one shouldn't, these results would be putting pressure on every model or theory or paradigm of bottom-up physics that goes beyond the Standard Model. In fact, SUSY theories are still among the "least constrained ones" among all paradigms that try to postulate some (motivated by something) new physics at low enough energy scales. That's the actual reason why the events cannot rationally justify the elimination or severe reduction of SUSY research as a percentage of hep-ph research.

If someone thinks that it's pointless to do physics without new guaranteed enough experimental discoveries and this kind of physics looks like a "problem" or "crisis" to him or her, he or she should probably better leave physics. Those who would be left are looking for more than just the superficial gloss and low-hanging fruits. The number of HEP experimenters and phenomenologists building their work on a wishful thinking of many collider discoveries in the near future is arguably too high, anyway. But there are other, more emotion-independent approaches to physics that are doing very well.

1. Back in the day, SciAm would have mocked Newton, too, for his highly speculative so-called physics and his nonsensical calculus.

2. I wonder if and how much such and similar trolling about fundamental physics in the Scientific American (and Nature) add to the reasons that funding of this field in particular (and for science generally) is going down the drain in the US ... :-/.

Can somebody powerful enough please ask the editor to make use of her possibility to stop including such topics in the Scientific American, as the magazin obviously cant report about such things in a correct and constructive way (they same holds for Nature) ...?

And while she is at it, maybe the title of the magazine should be changed too, as there is absolutely nothing "Scientific" about inflating such anti-science propagande and delusions of layman/philosophers/other outsiders (who have no clue about the scientific method among other things) on the cover or any part of the magazine...

3. Lubos, regarding your remarks about BICEP2 implications on GUT/SUSY have you seen this relevant paper by Lyth?

http://arxiv.org/abs/1403.7323

Check especially the last chapter. He advocates that BICEP2 do not favor GUT and low SUSY, on the contrary I would say.

What is your take on this?

Anyway about the Anti-Scientific American article, I totally agree with your remarks. The cover especially is despicable. These folks must understand that the tabloid mentality is not applicable in scientific press.

4. I'd say the present crisis in physics shouldn't be understated, but nor should the recent achievements. Many think there's a crisis because although we have a lot of pieces of the puzzle that we think may well go where we've placed them, we're short of pieces that we're certain about. Physics used to move forward by putting in pieces that were more certain.
You're right that "there can't really be any paradigm shift if the status quo continues. You either need some new experimental discoveries or some new theoretical discoveries for a paradigm shift." That's true. The crisis is because many think we need that.
The conceptual problems with the present picture keep getting overlooked, particularly the ones with time, and attempts are often made to move forward with mathematics alone. But conceptual progress is probably needed as well, and many feel the present kind of approach isn't getting us to where we need to go.

5. Dear Giotis, there are several claims he makes, some of them look utterly illogical to me, others are probably not right for subtle reasons.

Inflation never "required" SUSY, regardless of the scale. It's obvious that the de Sitter space breaks SUSY so as long as inflation occurs above the electroweak scale, and it was almost certain, SUSY would be broken during inflation - much more than it is broken now - which reduces its relevance. It doesn't have any implication for the present SUSY breaking scale. On the contrary, it would be more possible to derive something about high energies (inflation) from the low-energy observations (now): if SUSY is broken now, you can be sure that it was broken then.

The high energy density doesn't imply that cosmic strings etc. have to be around in abundance. So yes, shift symmetry etc. is probably more important for inflation to work, but that implies nothing in one way or another about the SUSY breaking scale.

Concerning the relics, the main thing that inflation actually suppresses - in a welcome way - is the concentration of magnetic monopoles. Magnetic monopoles per se are not really discussed in the paper at all. However, both magnetic monopoles and cosmic strings are being diluted in the inflationary epoch. This is true even if the Hubble scale is high. When the inflaton rolls down, the visible universe is already found in the same direction of the axion etc. so there are no topological relics there. So the only problem would be a recreation of these things during reheating. But during the reheating, from the moment when the expansion is only power-law, the energy density and temperature is already low enough to avoid them.

A paper I would more recommend about similar topics is e.g.

http://arxiv.org/abs/1404.2912

6. I don't know what to do with similar comments, Jonathan. They look internally inconsistent and completely vacuous to me.

You say that there is a crisis and one tries to understand what it even means. On one hand, you say that there is a crisis because we need to "go somewhere". But the main result of the experiments is that they don't want us go anywhere else! Instead, they are confirming the status quo.

So what you propose as the "preferred future of physics" contradicts the empirical evidence that you previously used to defend that there is a crisis in the first place. Equally seriously, what kind of a shift to new ideas so that we don't "overlook" your unspecified musings about time remains completely obscure. One can't do physics by making similar philosophically mysterious and physically vacuous claims. Moreover, the actual empirical data don't really force us to do any progress. It doesn't mean that we shouldn't work on conceptual progress - string theorists are doing it all the time, and extremely successfully - but it does mean that nothing like that is "required" by the actual status of the experimental results that are not "superfast" - BICEP2 sped things up a bit.

7. doctorIthinkI'mAphysicsHaterApr 16, 2014, 3:18:00 PM

Besides from this, http://www.physicsoverflow.org just officially lauched! Hooooora! Yes!

8. Well, "A Crisis In Physics" sells better than "Don't Worry, Be Happy. Physics is Doing Fine. Carry on"

9. Only for those who can't appreciate some fancy multi-voice music! ;-)

10. I'm not alone in thinking there's a crisis, if I was we wouldn't even be talking about it. And I said that the recent achievements in physics shouldn't be understated. I'm not advocating any "preferred future of physics", or putting down the progress that we make. Where we have testable ideas (and many wish there were more) it's great to be able to confirm them.
I'm just trying define what people mean when they talk about a crisis in physics, and throw in my own two cents as well. It's not about SUSY, it's more general. Many feel our present paradigm is short of a conceptual backbone, and that we're digging around the edges at present. Both Einstein and Wheeler talked repeatedly about finding an underlying conceptual basis for physics. But nowadays it seems to me we too often avoid the areas where the conceptual side is a bit frayed at the edges, as it is with time. But that's where the clues are. Incidentally, David Deutsch made a similar point in an interview recently. He said we should start by looking at what's wrong with our conceptual view of things, and try to sort that out first, and then try to do the mathematics. These are general points, but I think they get nearer the main issue than worries about SUSY, string theory, or anything else.

11. I know you're not alone. I didn't write you were alone. I wrote you are wrong. And the fact that you are not alone is sort of a necessary condition for someone to respond to the claim at all.

12. Philosophically, there might be a theory of all physics. But in terms of "operationally applicable definitions" I doubt that there could be a theory of all physics. The problem is to describe all possible physicists and all possible measuring devices. How could there be enough informational storage capacity for such a theory?

13. I'm not mudslinging, and I'm not the people whose views you're painting onto mine! I find the present era very exciting - because of a lot of things, but above all because we may be near a new paradigm. Both my work and the work I'm interested in looks in that kind of direction. If we don't find a new way forward of that kind, then myself and others were wasting our time to work on looking in that kind of direction. I'm not sure if you think we should be or not, but that's what we're doing anyway.

14. Only theory, that is, knowledge of natural laws, enables us to deduce the underlying phenomena from our sense impressions. When we claim that we can observe something new, we ought really to be saying that, although we are about to formulate new natural laws that do not agree with the old ones, we nevertheless assume that the existing
laws — covering the whole path from the phenomenon to our
consciousness—function in such a way that we can rely upon
them and hence speak of 'observations'...Physics and Beyond (pg67)(PDF)

The Magazine story strikes "at the core" and it is just a matter of understanding why one might react to it when in the long term, one realizes that they are firmly entrench within a process that is right. So not to worry. Calm the soul, Lubos. :)

15. We will never understand everything, David, but that doesn’t mean we will not understand the fundamental laws of physics that underly everything in the universe. For example, calculating the aerodynamic drag on a cruising airliner is beyond any imaginable computer yet the laws of fluid dynamics, which completely define the problem in precise detail, can be written on the back of a postage stamp. It’s only that the calculations become un wieldy that prevents an exact solution.
So it is with a theory of all physics (TOE or Theory of Everything). There will come a day when all the laws are known and all that remains is numerical calculation.

16. Lubosh: the article on "Supersymmetry and the Crisis in Physics" was co-authored by "Maria Spiropulu", not exactly a top notch physicist. Should read her articles with a grain of salt.

17. I find the present time very exciting and full of promise, for many reasons. But the main reason is that we may be near a new paradigm. If so, that would explain the doubts about the present paradigm. You've misinterpreted what I've said - I think we're doing well, not badly. (There are others who think we're doing badly, talk to them.) Nevertheless, I'm interested in finding ways forward, and whenever physics does find a way forward, there are always doubts about the existing paradigm beforehand. That's natural enough, and part of the process.

It's healthy to point out weaknesses in the present picture, and doesn't have to be negative. Arguably most of the major theories of the 20th century, not just SR, were attempts to reconcile discrepancies in existing theory. That's a part of how we move forward. If no-one pointed out the cracks in the picture, we'd make progress slower. It's positive to do that. So here's one. Few physicists have the courage to look at it closely. In SR with Minkowski spacetime, the future is fixed and pre-decided. In QM (unless in a hidden variables approach), the future is unavoidably unfixed, and undecided. It gets very unscientific and wishy-washy to assume both can be true at once, but many do.

18. Some fifty years ago physicists hoped to derive masses of elementary particles from some deep underlying principles. Are we today closer to that goal?

19. Ten years ago not many people would dare to speak about a crisis in high energy physics. Now the issue is on the front cover of a respectable scientific magazine. This means that it is becoming a common and acceptable knowledge. Decades of theoretical research leading nowhere or to untestable theories is the reason behind this frustration. The magazines are not to blame for critisizing this dead end. Theoretical physicists are, because they have cut their ties with reality and they are building models out of thin air. Only stubborn people like you Motl, who have spent their lives hunting ghosts, are unable to admit their failure.

20. I think that Maria is a great (experimental) physicist!

21. Vladimir KalitvianskiApr 16, 2014, 9:11:00 PM

Supersymmetry is no even reading mind of Nature (which is difficult by itself). https://www.youtube.com/watch?v=zmn5BH_ga8k

22. Of course we are doing very well, Jonathan, but your use of the word “crisis” is inapt at best and to speak of “weaknesses in the present picture” is even worse. Even more ridiculous is your insulting assertion that “few physicists have the courage to, look at it (cracks in the picture) closely. Nothing, absolutely nothing could be farther from the truth.
And do you actually believe that many (physicists) assume both that the future is determinate and that it is not?

23. Jonathan, of course QM and SR are true at once; the unified theoretical framework is called QFT:-)

24. Please stop pretending that you are working at or even begin to understand the frontiers of theoretical physics, Jonathan.

25. Dear Professor,

On a side note can you tell me if this news is correct?

http://msutoday.msu.edu/news/2014/plugging-the-hole-in-hawkings-black-hole-theory-1/

26. If you write to me, I'll send you my book when it comes out, "The unsolved puzzle: motion through time in 21st century physics", and you can decide for yourself, without having to guess. I've taken the four or five main avenues on the time issue, and examined them closely. I think this is valuable, because I think we need to take this head on, instead of sweeping it to one side. And as I said, necessity is the mother of invention. New paradigms, like SR, arise out of inconsistencies in the previous picture. But of course, you have to admit they exist.

27. Particle physics is short of ideas currently with or without string theory. One may call it a crisis, but astrophysics on the other hand is super exciting. So on average physics is doing fine.

28. You certainly know that classical physics (even for large velocities ...) is nothing but the classical limit of QM, no? There is absolutely no contradiction between the (classical) predictions of SR and QM.

Pointing out crack and weaknesses is a good thing, but people who claim to point out cracks and weaknesses should confuse them with their personal lack of knowledge or understanding, which happens way too often ...

29. Do we have an aggressive anti-science spam attack in this thread ...?

30. Dear Bill, is *astrophysics* today or astrophysics in general superexciting? I must be living in a different galaxy. Didn't you confuse astrophysics with *cosmology*?

You say that particle physics is "short of ideas" - relatively to what? How do you measure the right amount of ideas? What is the threshold of the "number of ideas" above which a science discipline becomes "long of ideas"? What do all these hostile but otherwise meaningless proclamations mean - except for nothing?

What is particularly ironic is that the same people who criticize particle physics for being "short of ideas" also criticize it for having "too many ideas".

31. Apologies, Dilaton, I just decided to approve everything so that we see how such threads end up without moderation that judges the content and not just the form.

32. I'm absolutely sure that QM and SR are true at once, and I never said they weren't. The one untested (and untestable) area of our present picture is Minkowski spacetime, and that is exactly where all the conceptual problems have arisen. They're often ignored because they're not mathematical problems. I think the assumptions we take on trust about the time dimension from Minkowski must be wrong somewhere.

33. Aah ok ... ;-)

34. If a field whose goal is to uncover the nature of reality were to have discovered the ultimate nature of reality, it would have in some sense reached its destination. This hasn't happened yet.

So, what exactly is the crisis?

35. I'm not a physicist, but you seem way too enamoried of the concept of 'paradigm' to be one, either. Social scientists and journalists are super impressed by this broad concept because their vague ideas about their vague topics sound impressive when couched as a major viewpoint, e.g. 'pyschoanalysis paradigm'. much of social science has never escaped the realm of opinion and style, it doesn't get beyond those. Physics is not about opinions or styles, and the laws and equations are what they are, regardless of whether some journalist talks about a paradigm (or a crisis)

36. It is easy to measure the number of ideas in a branch of physics(or the lack of it). You only need to watch how often they refer to the anthropic principle. If you think about it, it really is a good measure, because you only invoke the anthropic reasoning as a last resort.

So no, physics between 1910 and 1920 was not really short of ideas in this sense.

As far as the physics haters are concerned(where I apparently belong), I don't know about the others, but I never said particle physicists have too many ideas I can't even imagine that being a problem.

Currently I am afraid that when I get there I will find string theory too *arbitrary*, but I never articulated such concerns either.

37. Is it possible to have a unification without supersymmetry?

38. There's a crisis in physics in these magazines every six months. It's totally surprising that the vacuum has not decayed to a jelly state by now.

39. The great phenomenological findings from astrophysics/cosmology are meaningless without a theoretical physics framework to interpret them.

Dark matter was discovered in the 1930s and the 1970s. In the 1930s, nobody cared about dark matter and it was dismissed, because particle physics was in a then relatively primitive state and thus the discovery could not be integrated into any reasonable theoretical framework.

40. I'm absolutely sure that QM and SR are true at once, and I never said they weren't. They've been well tested. The one untested (and perhaps untestable) area of our present picture is Minkowski spacetime, and that is exactly where all the conceptual problems have arisen. They're often ignored because they're not mathematical problems. I think the assumptions we take on trust about the time dimension from Minkowski must be wrong somewhere. Even a minor error could remove block time, the contradiction about the future I mentioned, and other contradictions.

41. The word effectively has a different meaning in physics.

42. Physics, yes, but what kind of physics? Not all of astrophysics depends on further progress in fundamental physics. Take the undeniably hot topic of extrasolar planets. The ultimate motivation for this interest probably has to do with our desire to find life in the Universe more than any purely scientific reason. Dealing with all the data on extrasolar planets will still require plenty of physics, but of a more established kind. As far as physics goes, it's more of an area for applied physics, both in terms of improving instrumentation and in terms of interpreting the data.

So, while the areas of astrophysics that overlap with fundamental physics are undoubtedly fascinating, the field of astrophysics would still remain exciting to many people even if fundamental physics research were to suddenly stop for some reason.

43. Based on your screen name, I assume that you're a deliberate troll.

44. As long as you do not have a theory that describes two electrons having electromagnetic and gravitation interaction at the same time in a CLEAR fashion then TOE is hopeless and that means physics is in deep trouble, SUSY or no SUSY.

45. What happened to down voting? I noticed the down votes having disappeared a couple weeks back.

46. Scientific American has been getting worse over the years. If you think they screw up physics, look what they do with less sciences. They are complete shills for the global warming apocalyptic cult - the latest issue has 3 articles in that regard, including one really bad one by Michael Mann.

SA has long been owned by leftist/progressive sorts. In the '80s, it was full of articles mocking every bit of the strategy the US used to defeat the USSR. Every new weapon system was shown to not be workable (never mind how well they worked a few years later in Iraq). Missile defense was impossible, impractical, and not only that, positively evil.

They ended their outstanding Amateur Scientist section, after they discovered that their hire for the new editor was a creationist (Forest Mims). Now, before you jump up and down, the guy was perfectly competent at the sort of things they hired him for, and a good observational scientist (field biology, etc). But, that didn't matter. They couldn't have such a person at their magazine - even though a lot of their own attitudes were just as unscientific.

I still subscribe, but the value keeps dropping.

47. I picked the screen name when there was fear of a widespread swine flu epidemic and haven't been able to decide whether to change it or keep it.

So, one more time, and like before, with no intention of trolling. Not all of astrophysics is there to "serve" fundamental physics, and these other parts of it will happily chug along regardless of the state of fundamental physics research. This is not an "attack" on fundamental physics, just a comment on the nature of astrophysics as a field. It is too narrow a view of astrophysics as a field to think otherwise.

48. It's true that parts of astrophysics are doing well without input from fundamental theoretical physics, as is the case with palaeontology, architecture, music theory, etc. all of which are independently important and interesting and I hope that people continue to make progress.

Some parts of astrophysics are intermingled with theoretical physics, and some parts are not.

If you genuinely believe that there is no progress being made in theoretical physics, the first answer that you should consider is that you're simply not aware of the progress that's being made, because it's not being communicated properly or because you don't understand it.

49. Why would I think that no progress is being made in fundamental physics? I never made such a claim. I was referring only to the part of Bill Bogus' post where he claimed that astrophysics was in a "super exciting" state, which it is - people talk about its being in a golden age for a reason.

I don't, however, think that it is in such a happy state only because of the discoveries/observations related to fundamental physics problems, that was my beef with your comment on his, if I understood it correctly. The comparison to architecture is cute, but a bit overdone. Planet formation theories, to repeat my earlier example, require tools that are undeniably physics tools, albeit rather established ones. And it is hardly the only example in astrophysics.

50. Swine,

My apologies, I've confused the content of your posts with those of Bill's.

51. Many people seek fame and money; And to claim a crisis has proved to be one of the most sure ways of attracting attention and to get to bask in the lime-light, sell "copies", or score higher media ratings that increase the income from advertisers. %-[

52. Dear Lubos,
i'm a fellow physicist from Italy. A country, as you often points out on your blog, that is on constant attack against science (it's really sad).
I think there is a crisis in physics indeed, but it's not the kind of false problems that SciAm or other misleading people (just to name some in increasing order of mental confusion: Turok, Woit, Smolin...) like to write about.
The problem with physics is double-face: on one side we have this increasing and worrying attitude towards modern physics. They really don't see that making a temporal cut in any epoch since Galileo's time we will find a period with a bit of confusion, conjectures, wrong and right theories, proposed solutions and so on... So, physics in XXI cen. is just the usual business of finding right explanatons for natural phenomena. Moreover, the pace of physics is not decreasing at all. In the last 20/30 years we had plenty of discoveries and new insights in every field of physics. Even in particle physics. These people seems do not understand that science is a win-win game: if you find SUSY that's great, if you don't find SUSY, that's great too, because this fact will nonetheless sharpen our understanding of the world.
The second face of the problem in physics is of course entangled to the first one. These continuous anti-physics tirades are producing more and more detachment. I'm actually observing a growing distrust of physics and science in general by laymen. This, in the long run, will surely produce the nasty effect of fewer and fewer funds towards real science. We have already witnessed in these years billions of euros and dollars going in the direction of IPCC's horoscopes and other similiar unscientific (and mostly political) issues.
These are, in my humble opinion, the real problems that physics is facing today. In a slogan: physics is fine, people are fucked.

53. Marcel van VelzenApr 17, 2014, 9:58:00 AM

Dear Lubos,

On a more positive note: The film "particle fever" was brought-casted
last evening/night on Dutch television. I wanted to watch just half an
hour and see the rest today but I (and my youngest son) watched all of
it until almost one o'clock. What a brilliant movie. It is very balanced
between theory and experiment; standard model physics and new models,
personal life and background, hope, disappointment and victory, hard
facts and philosophical considerations. Both Savas Dimopoulos and Nima
Arkani Hamed are doing an excellent (and even witty) job in explaining
what drives particle physicists these days. Excellent!

54. Dear Motl
Your blog is very high quality physics (and intellectual) entertainment.
Thanks
Abraham

55. To clarify that, both meanings of the word paradigm are about a general view of things, or a general way of seeing something. But the word paradigm in say the social sciences doesn't necessarily have anything underneath it, and can be a trend, as you say. However, in physics, what we call a paradigm means a general view that becomes the standard view. And for that to happen, because physics is not a vague discipline, there has to be major supporting evidence for it, both theoretical and experimental evidence. I suggest you read Thomas Kuhn's "The structure of scientific revolutions", to understand what you have posted about properly.

56. There are two problems here: Is there a crisis in physics because of lack of new experimental signatures?

My answer is NO. I have lived through Regge poles describing data, vector meson dominance, the eightfold way, SU(3)xSU(2)xU(1) as they appeared and dominated the particle physics analysis. Surprises were coming as energy scales grew. I expect there will be surprises in the new run, and also expect that a lepton TEV collider might do a much better job at getting signals of new physics.

As you say "if it is not supersymmetry it will be something else".

The second problem comes from a, deluded or not, fifth phalanx, as we call in Greek insidious support to a "crisis" shout from physicists who should know better. Yes HEP research is expensive but is a small percentage of the money spent on arms by the big countries. Still there are a number of feeders at the public trough who eye the HEP money enviously and will pick up any negative reports that support their grab for the money. Therein is the real problem imo.

57. alejandro riveroApr 17, 2014, 9:09:00 PM

People has been looking at Planck scale much more time that at TeV/GeV/MeV scales, which is where the action is. That is the crisis. Nobody has worried about explaining why the yukawa of the top is 0.99 (actually, 1-sigma compatible with 1.00) not any other hint from low energy because everything, as de Rujula drew once, was going to come from the Heights.

58. Exactly, there are scornful and hateful people who begrudge HEP and fundamental physics every cent it gets :-/.

Their goal to get rid of this part of research will get strong tailwinds due to the scientifically not justified hapless screams about crises, we dont understand a thing, etc and other negative talks, in the popular and sometimes even in the not so popular chanels ...

In the US the negative impact of these anti science movements can already be observed with a significance of much more than 5 sigma ...

59. I just need to clarify that, that’s all. Both meanings of the word paradigm are often about a general approach or model, or a general way of seeing something. But the word paradigm in, say, the social sciences doesn’t necessarily have anything underneath it, and can be a trend, as you say.

However, in physics, what we call a paradigm means a view or model that becomes the standard one. And for that to happen, because physics is not a vague
discipline, there has to be major supporting evidence for it, both theoretical
and experimental evidence. So that’s what was being talked about. I suggest you
read Thomas Kuhn’s book “The structure of scientific revolutions”, who coined
the phrase ‘paradigm-shift’, and didn’t see the word paradigm as applicable to
the social sciences.

60. Marcel van VelzenApr 17, 2014, 11:52:00 PM

If you do all the canonical normalizations right, the Yukawa coupling of
the top quark is not 1 but about 0.7. See Weinberg's "The quantum
theory of Fields 2", formulas 21.3.21 and 21.3.31: Yukawa
coupling=Mtop/v, with v the vacuum expectation value of the Higgs field. Still pretty close to 1!

61. What could possibly be unclear about the simultaneous gravitational and electromagnetic interaction between two electrons? You are spouting gibberish.