Thermodynamics and Statistics

Abstract

Classical thermodynamics can be derived from a set of axioms, but this chapter starts out from the statistical nature of quantum theory, thereby giving a meaning to these axioms. The binomial, Poisson, Gauss, and Maxwell distributions are compared with their implications for average values, correlations, and uncertainties. A basic issue is the minimum size of the phase space element. While classical differential equations are invariant under time reversal, the notion of entropy breaks this symmetry. We derive the relaxation-time approximation, the Liouville, collision-free Boltzmann, Langevin, and Fokker–Planck equations, and the fluctuation–dissipation theorem. The entropy theorem implies thermal, mechanical, and chemical equilibrium conditions using micro-canonical, canonical, grand-canonical, and other ensembles, with applications to gases, mixtures, Fermi gases, photons, and lattice vibrations. Phase transitions of first and second order and critical behavior in (real) gases, para- and ferromagnets, and also Bose–Einstein condensation are all reviewed. There is a list of 42 problems.

Appendices

Problems

Problem 6.1

Legend tells us that the inventor of chess asked for $S=\sum ^{63}_{z=0}2^z$ grains of rice as a wage: one grain on the first square, two on the second, and twice as many on each subsequent square. Compare the sum S for all the squares with the Loschmidt number $N_\text {L}\approx 6\times 10^{23}$. How often can the surface of the Earth be covered with S grains, if 10 of them are equivalent to 1 square centimeter? By the way, 29% of the surface of the Earth is covered by land.    (3 P)

Problem 6.2

Justify Stirling’s formula $n!=(n/{\text {e}})^n\;\sqrt{2\pi n}$ with the help of the equation $n!=\int ^\infty _0 x^n\;\exp (-x)\;{\text {d}}x$, using a power series expansion of $n\ln x-x$ about the maximum and also by comparing with $\ln (n!)$, $n\ln (n/{\text {e}})$, and $n\ln (n/{\text {e}})+\frac{1}{2}\ln (2\pi n)$ for $n=5$, 10, and 50.    (9 P)

Problem 6.3

Draw the binomial distribution $\rho _z=\left( {\begin{array}{c}Z\\ z\end{array}}\right) \;p^z(1-p)^{Z-z}$ when $Z=10$ for $p=0.5$ and $p=0.1$. Compare this with the associated Gauss distribution (equal to $\langle z\rangle$ and $\Delta z$) and for $p=0.1$ with the associated Poisson distribution. Note that the Gauss and Poisson distributions also assign values for $z>10$, but which we do not want to consider. For comparisons, set up tables with three digits after the decimal point, no drawings.    (8 P)

Problem 6.4

From the binomial distribution for $Z\gg 1$, derive the Gauss distribution if the probabilities p and $q=1-p$ are not too small compared to one.

Hint: Here it is useful to investigate the properties of the binomial distribution near its maximum and let $\rho$ depend continuously on z.    (8 P)

Problem 6.5

How high is the probability for z decays in 10 seconds in a radioactive source with an activity of 0.4 Bq? Give in particular the values $\rho (z)$ for $z=0$ to 10 with two digits after the decimal point.    (6 P)

Problem 6.6

Which probability distribution $\{\rho _z\}$ delivers the highest information measure $I=-\sum _{z=1}^Z\rho _z\,\text {lb}\,\rho _z$?

Hint: Note the constraint $\sum _{z=1}^Z\rho _z=1$.

How does I change if initially $Z_1$ states are occupied with equal probability and then $Z_2<Z_1$? Freezing of degrees of freedom: Determine $\Delta I$ for $Z_1=10$ and $Z_2=2$. For two possibilities, I may be written as a function of just $p=\rho (z_1)$. Set up a table of values with the step width 0.05.    (6 P)

Problem 6.7

In phase space, every linear harmonic oscillation proceeds along an ellipse. How does the area of this ellipse depend on the energy and oscillation period? By how much do the areas of the ellipses of two oscillators differ when their energies differ by $\hbar \omega$? Determine the probability density $\rho (x)$ for a given oscillation amplitude $x_0$ and equally distributed phases $\varphi$.

Hint: Thus we may set $x=x_0\sin (\omega t\!+\!\varphi )$. Actually, the probability density should be taken at time t. Why is this unnecessary here?    (7 P)

Problem 6.8

A molecule in a gas travels equal distances l between collisions with other molecules. We assume that the molecules are of the same kind, but always at rest, a useful simplification which does not falsify the result. Here all directions occur with equal probability. Determine the average square of the distance from the initial point after n elastic collisions, and express the result as a function of time.    (4 P)

Problem 6.9

Does $\rho (t,\mathbf {r})=\sqrt{4\pi Dt}^{\;-3}\,\exp (-r^2/4Dt)$ solve the diffusion equation $\partial \rho /\partial t=D\Delta \rho$, and does it obey the initial condition $\rho (0,\mathbf {r})=\delta (\mathbf {r})$? What is the time dependence of $\langle r^2\rangle$? Compare with Problem 6.8. How do the solutions $\rho (t,\mathbf {r})$ read in one and two dimensions?    (9 P)

Problem 6.10

Consider N interaction-free molecules each of which is equally probable in any of two equal sections of a container. What is the probability for all N molecules to be in just one of the sections? If each of the possibilities since the existence of the world ($2\times 10^{10}$) has occurred corresponding to its probability, how long have 100 molecules (very, very few for macroscopic processes!) been in one section?    (2 P)

Problem 6.11

Given the Maxwell distribution

$$ \rho \,(v)=4\pi \,v^2(2\pi kT/m)^{-3/2}\;\exp (-mv^2\!/2kT)\;, $$

determine the most frequent and the average velocities ($\widehat{v}$, $\langle v\rangle$), kinetic energies ($\widehat{E}$, $\langle E\rangle$), and de Broglie wavelengths ($\widehat{\lambda }$, $\langle \lambda \rangle$).

Hint:

$$\begin{aligned} \int ^\infty _0\exp (-\alpha x^2)\;{\text {d}}x= & {} \frac{1}{2}\sqrt{\frac{\pi }{\alpha }}\;,\\ \int ^\infty _0x^{2n}\exp (-\alpha x^2)\;{\text {d}}x= & {} (-)^n\frac{\partial ^n}{\partial \alpha ^n}\int ^\infty _0\exp (-\alpha x^2)\;{\text {d}}x =\frac{(2n-1)!!}{2^{n+1}\alpha ^n} \sqrt{\frac{\pi }{\alpha }}\;,\\ \int ^\infty _0x^{2n+1}\exp (-\alpha x^2)\,{\text {d}}x= & {} \frac{1}{2}\int ^\infty _0y^n\exp (-\alpha y)\,{\text {d}}y=\frac{n!}{2\alpha ^{n+1}}\;. \end{aligned}$$

The first integral is half as large as $\int ^\infty _{-\infty }$ and the latter equal to the square root of the surface integral

$$ \iint ^\infty _{-\infty } \exp \{-\alpha (x^2+y^2)\}\,{\text {d}}x \,{\text {d}}y =2\pi \int ^\infty _0\exp (-\alpha r^2)\,r\,{\text {d}}r =\pi \int ^\infty _0{\text {e}}^{-\alpha x}{\text {d}}x\,. $$

   (8 P)

Problem 6.12

Consider the 1D diffusion equation $\partial y/\partial t=D\;\partial ^2y/\partial x^2$ with the boundary condition $y(t,0)=c(0)\exp (-\text {i}\omega t)$. Which differential equation follows for c(x), and what are its physical solutions for $x>0$? (Example: seasonal ground temperature.)    (3 P)

Problem 6.13

Under what circumstances do the Maxwell equations yield a diffusion equation for the electric field strength? How large is the diffusion constant under such circumstances?    (3 P)

Problem 6.14

For a molecular beam, all velocities $\mathbf {v}$ outside of a small solid angle ${\text {d}}\Omega$ around the beam direction are suppressed. How large is the number of suppressed molecules with velocities between v and $v+{\text {d}}v$ per unit time and unit area? Determine the most frequent and the average velocity in the beam.    (4 P)

Problem 6.15

According to quantum theory, the phase space cells cover the area h. Therefore, according to Problem 6.7, the number of states of one linear oscillator up to the highest excitation energy E is equal to $\Omega (E,1)=E/\hbar \omega +1=n+1$, with the oscillator quantum number n. Determine $\Omega (E,2)$ for distinguishable oscillators and then $\Omega (E,N)$ by counting. Simplify the result for the case $n\gg N$. Is the density of states for this system equal to $\frac{1}{N!}E^{N-1}\;(\hbar \omega )^{-N}$?

Hint: The binomial coefficients for natural m and arbitrary x are given by

$$\begin{aligned} \left( {\begin{array}{c}x\\ m\end{array}}\right) =\frac{x\cdot (x-1)\cdots (x-m+1)}{m!}= \frac{x-m+1}{m}\;\left( {\begin{array}{c}x\\ m-1\end{array}}\right) \;. \end{aligned}$$

Consequently,

$$\begin{aligned} \left( {\begin{array}{c}x\\ 1\end{array}}\right) =x=x\left( {\begin{array}{c}x\\ 0\end{array}}\right) \;,\qquad \left( {\begin{array}{c}m\\ m\end{array}}\right) =1=\frac{1}{m} \left( {\begin{array}{c}m\\ m-1\end{array}}\right) \,,\quad \text {and for }n<m\,,\quad \left( {\begin{array}{c}n\\ m\end{array}}\right) =0\;. \end{aligned}$$

In addition,

$$\begin{aligned} \left( {\begin{array}{c}x+1\\ m\end{array}}\right) =\left( {\begin{array}{c}x\\ m\end{array}}\right) +\left( {\begin{array}{c}x\\ m-1\end{array}}\right) \;,\quad \text {and hence}\quad \left( {\begin{array}{c}n+1\\ m+1\end{array}}\right) =\sum ^{n-m}_{k=0}\left( {\begin{array}{c}n-k\\ m\end{array}}\right) \;. \end{aligned}$$

   (6 P)

Problem 6.16

From the expression found for $\Omega (E,N)$ in Problem 6.15, determine the canonical partition function and hence the average energy $\langle E\rangle$ and the squared relative fluctuation $(\Delta E/\langle E\rangle )^2$.    (4 P)

Problem 6.17

The energy of N non-interacting spin-$\frac{1}{2}$ particles with magnetic moments $\mathbf {\mu }$ in the magnetic field is $E=(n_{\downarrow \uparrow }-n_{\uparrow \uparrow })\;\mu B$. What is the micro-canonical partition function of this system?    (4 P)

Problem 6.18

Take the result of the last problem as a binomial distribution (with the energy as state variable), and approximate it by the Gauss distribution for $\mu B\ll {\text {d}}E\ll E$. Thereby determine the entropy. How does the entropy differ from the one found in Problem 6.17, obtained with the Stirling formula for $E\ll N\mu B$?    (6 P)

Problem 6.19

Determine, as for the equidistribution law, $\langle p_n\dot{x}^m\rangle$ and $\langle x^m\dot{p}_n\rangle$ for canonical ensembles of particles which are enclosed between impenetrable walls. Why are these considerations not also valid for unbound particles?    (4 P)

Problem 6.20

For an N-particle system, the expression $\sum ^N_{i=1}\mathbf {r}_i\cdot \mathbf {F}_i$ is called the virial of the force. What follows for its expectation value? Compare the result with $\langle E_\text {kin}\rangle =N\;\frac{m}{2} \langle \dot{\mathbf {x}}\cdot \dot{\mathbf {x}}\rangle$ and with the virial theorem of classical mechanics. Note that this holds for the mean value over the time (!), and in fact for “quasi-periodic” systems, i.e., x and p always have to stay finite.    (5 P)

Problem 6.21

Consider the 1D diffusion equation $\partial y/\partial t=D\;\partial ^2y/\partial x^2$. How do its solutions read with the initial condition $y(0,x)=f(x)$ instead of the boundary condition of Problem 6.12?    (2 P)

Problem 6.22

The gas pressure p on the walls can be determined from the momentum change due to the elastic collision of the molecules. Determine the pressure as a function of the average energy of the individual molecules. Here the same assumptions are made as for the derivation of the Boltzmann equation. Do we need the Maxwell distribution? What follows for $\langle E\rangle$ if the ideal gas equation $pV=NkT$ holds?    (6 P)

Problem 6.23

In a galvanometer, a quartz fiber with the torque $\delta = 10^{-13}$ J supports a plane mirror. How large is the directional uncertainty at $20\,^\circ \text {C}$ from the Brownian motion of the air molecules? How much does a reflected light beam fluctuate on a target scale at 1 m distance?    (3 P)

Problem 6.24

For an ideal monatomic gas, $pV^{5/3}$ is a constant for isentropic processes. How much does the internal energy U change if the volume increases from $V_0$ to V? Does U increase or decrease?    (3 P)

Problem 6.25

Consider a cycle in an (S, T) diagram. What area corresponds to the usable work and what area to the heat energy input? Consider a heat engine with the heat input $Q_+=T_+\Delta S_1$ at the temperature $T_+$ and $Q_0=T_0\Delta S_2$ at $T_0<T_+$, as well as heat output $Q_-=T_-\;(\Delta S_1+\Delta S_2)$ at $T_-<T_0$. Determine the efficiency $\eta (Q_+,Q_0,Q_-)$ and compare it with the efficiency of an ideal Carnot process ($\eta _\text {C}$ with $Q_0=0$). Express the result as a function of $\eta _\text {C}$, $Q_0/Q_+$, and $T_0/T_+$. Determine a least upper bound for the efficiency of a cycle process with heat reservoirs at several input and output temperatures.    (9 P)

Problem 6.26

Why do we have to do work to pump heat from a cold to a hot medium? Investigate this with an ideal cycle. Under ideal constraints, let the work A be necessary in order to keep a house at the temperature $T_+$ inside, while the temperature outside is $T_-$. How are these three quantities connected with the heat loss $Q_+$? How is the input heat $Q_+'$ in an ideal power plant related to the heat loss $Q_+$ considered above if it works between the temperatures $T_+'$ and $T_-'$? Neglect the losses in the power plant that delivers the electric energy. Take as an example $T_+'=800\,^\circ \text {C}$, $T_+=20\,^\circ \text {C}$, and $T_-=T_-'=0\,^\circ \text {C}$.    (8 P)

Problem 6.27

Determine the functional determinant

$$\begin{aligned} \frac{\partial (S,T)}{\partial (V,p)}= \left( \frac{\partial S}{\partial V}\right) _p \left( \frac{\partial T}{\partial p}\right) _V- \left( \frac{\partial S}{\partial p}\right) _V \left( \frac{\partial T}{\partial V}\right) _p\;. \end{aligned}$$

   (2 P)

Problem 6.28

Express the derivatives of S with respect to T, V, and p, with the other parameters kept fixed, in terms of the thermal coefficients and V and T. Express the derivatives of T with respect to S, V, and p in terms of the quantities above. Express $(\partial F/\partial T)_p$ and $(\partial G/\partial T)_V$ in terms of these quantities.    (6 P)

Problem 6.29

Are $(\partial ^2U/\partial S^2)_V$, $(\partial ^2U/\partial V^2)_S$, $(\partial ^2G/\partial T^2)_p$, and $(\partial ^2G/\partial p^2)_T$ always positive?    (4 P)

Problem 6.30

If a charge ${\text {d}}q$ is inserted isothermally and isochorically into a reversibly working galvanic element at the open circuit voltage $\Phi$, the work $\updelta A=\Phi \,{\text {d}}q$ is done. How does its internal energy change for given $\Phi (T)$?

Hint: Note the integrability condition for the free energy F. In addition, we should have $\updelta A=\varphi {\text {d}}Q$, if upper-case letters always stand for extensive quantities and lower-case letters for intensive quantities.    (4 P)

Problem 6.31

What vapor pressure p(T) is obtained from the Clausius–Clapeyron equation if we assume a constant transition heat Q, neglecting the volume of the liquid compared to the volume of the gas, and using the equation $pV=NkT$ for an ideal gas?    (4 P)

Problem 6.32

One liter of water at $20\,^\circ \text {C}$ and normal pressure (1013 hPa) is subject to a pressure twenty times the normal pressure. Here the compressibility is 0.5/GPa on average and the expansion coefficient $2\times 10^{-4}$/K. Determine $V/V_0$ as a function of p and $p_0$ (give values in numbers as well). How much work is necessary for the change of state? By how much does the internal energy change?    (6 P)

Problem 6.33

At the freezing temperature, ice has the density 0.918 g/cm$^3$ and water the density 0.99984 g/cm$^3$. An energy of 6.007 kJ/mole is needed to melt ice. How large are the discontinuities in the four thermodynamic potentials for this phase transition (relative to one mole)?    (4 P)

Problem 6.34

What is the connection between $(\partial U/\partial V)_T$ and $(\partial \frac{p}{T}/\partial T)_V$? Can $(\partial C_V/\partial V)_T$ be uniquely determined for a given thermal equation of state? Transfer the results to the enthalpy and $C_p$.    (6 P)

Problem 6.35

For a given heat capacity $C_V(T,V)$ and thermal equation of state, is the entropy uniquely defined? Can we then also determine the thermodynamic potentials?    (4 P)

Problem 6.36

From thermal coefficients for ideal gases, derive the relation

$$ pV^{\kappa _T/\kappa _S}=\text {const.}\;, $$

for isentropic processes. Determine V(T) and p(T) for adiabatic changes in ideal gases. How does the sound velocity c in an ideal gas depend on T, and what is obtained for nitrogen at 290 K?    (6 P)

Problem 6.37

For a mole of $^4$He at 1 bar and 290 K, determine the thermal de Broglie wavelength $\lambda$, the fugacity $\exp (\mu /kT)$, the free enthalpy (in J), and the entropy (in J/K). Here, helium may be taken as an ideal gas.    (4 P)

Problem 6.38

How is the thermal equation of state for ideal monatomic gases to be modified in order to account to first order for the difference in $\ln Z_\text {GC}$ between bosons and fermions?

Hint: We may expand pV/kT in powers of the fugacity and express this in terms of N, V, and $\lambda$.

Compare the pressures of the Bose and Fermi gases with that of a classical gas.    (8 P)

Problem 6.39

How do the pressure and temperature of the air depend on the height for constant gravitational acceleration if heat conduction is negligible compared to convection and therefore each mass element keeps its entropy? This is more realistic than the assumption of constant temperature.    (2 P)

Problem 6.40

Consider the heating of a house as an isobaric–isochoric situation: the air expands with increasing temperature and escapes through leakages. Assuming an ideal gas, how does the number of molecules in the house change, and how does the internal energy change, assuming that there are no internal excitations of the molecules? Does the entropy increase or decrease. Or is this clear anyway from the entropy law? (Heating is not an energy problem, but an entropy problem!)    (5 P)

Problem 6.41

To extend a surface by ${\text {d}}A$, work $\updelta W=\sigma \, {\text {d}}A$ has to be done against the attraction between the molecules, where $\sigma$ is the surface tension. What sign does $(\partial \sigma /\partial T)_A$ have? How does the free energy change for an isothermal surface (without volume change) and how does the internal energy change? How much heat is involved in an isothermal surface extension assuming that $\sigma (T,A)$ is given?    (6 P)

Fig. 6.31

Diesel cycle. Idealized cycle from 1 to 2 and from 3 to 4 along isentropic (adiabatic) curves of an ideal gas, between either isobaric ($2\rightarrow 3$) or isochoric ($4\rightarrow 1$) curves. Contrast with twice isochoric for the Otto cycle and twice isobaric for the Joule cycle (gas turbine)

Problem 6.42

For four-stroke engines (suction, compression, combustion, ejection), only two cycles are assumed to be idealized. For example, Fig. 6.31 shows the diesel cycle. Note that diesel engines are “compression–ignition engines”: the fuel burns at approximately constant pressure. Which two cycles are related to the diesel cycle (why?), and which path is taken by the one and the other in Fig. 6.31? What is the efficiency of the idealized diesel engine as a function of the compression $\text {K}=V_1/V_2$ and expansion $\text {E}=V_4/V_3$, assuming a single ideal diatomic gas, i.e., assuming the air to be pure nitrogen? Note that, clearly, $\text {K}>\text {E}>1$. Begin by expressing $Q_\pm$ in terms of the relevant temperatures. The compression depends on the construction, but the expansion does not. It is determined by the “heat of combustion” (combustion enthalpy). Determine the ratio K/E of the enthalpies.    (9 P)

List of Symbols

We stick closely to the recommendations of the International Union of Pure and Applied Physics (IUPAP) and the Deutsches Institut für Normung (DIN). These are listed in Symbole, Einheiten und Nomenklatur in der Physik (Physik-Verlag, Weinheim 1980) and are marked here with an asterisk. However, one and the same symbol may represent different quantities in different branches of physics. Therefore, we have to divide the list of symbols into different parts (Table 6.3).

Table 6.3 Symbols used in thermodynamics and statistics