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Ludwig BoltzmannThe Man Who Trusted Atoms$

Carlo Cercignani

Print publication date: 2006

Print ISBN-13: 9780198570646

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198570646.001.0001

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(p.283) Appendix 7.1 The canonical distribution for equilibrium states

(p.283) Appendix 7.1 The canonical distribution for equilibrium states

Source:
Ludwig Boltzmann
Publisher:
Oxford University Press

The basic problem of determining the distribution of microscopic states in a macroscopic equilibrium state can evidently be solved only if we restrict the systems under consideration. If one makes the assumption that no region of phase space is privileged, i.e. if the trajectories of the point representing the mechanical state of the system pass the same number of times (in a very long time interval) in each region having a given volume, then it is reasonable to think that the N-particle distribution function P (see Chapter 4) is constant. Of course, the existence of constants of the motion modifies this conclusion; if one makes the assumption (called ergodic in modern terminology) that there is only one important integral of the motion, the total energy, then the distribution function will be constant only on each hypersurface of constant energy. Gibbs considers systems in which this is more or less explicitly assumed. The distribution that is obtained is the microcanonical distribution, which correctly describes an isolated system with an assigned energy. In principle one can develop thermodynamics starting from this distribution, but the treatment is made complicated by the necessity of passing to the limit when N → ∞ in each expression. It is then more convenient to pass to the so-called canonical distribution, which may be obtained for the subsystems with a finite number of degrees of freedom, when the same number is allowed to go to infinity for the entire system.

The canonical system can be easily obtained from the microcanonical one by imagining the system we are interested in as weakly coupled with a thermostat which may be assumed to coincide with a perfect gas. Weak coupling means that the interaction is sufficient to reach thermal. equilibrium, but the corresponding interaction energy is negligible with respect to the thermal one. The circumstance that the system we are interested in, S, is coupled to a thermostat T means that the latter has a number of degrees of freedom much larger than the s degrees of freedom possessed by S, in such a way that the number of degrees of freedom of T can be assumed to tend to infinity.

In fact, let H s and H T be the Hamiltonian functions of the system and of the thermostat, and E be the total energy (system + thermostat); then the distribution function P for the enlarged system, including the thermostat, is given by:

(A7.1) Appendix 7.1 The canonical distribution for equilibrium states
where, if V(E) is the volume of the region in phase space defined by H s + H TE, we have let A(E) = dV/dE (an invariant measure of the area of the hypersurface bounding the region under consideration). We can also compute the areas A s(∊) and A T(E − ∊) corresponding to the boundaries of the two regions H s ≤ ∊, H TE − ∊: it is clear that there must he a simple relation between these two areas and. A(E). To find it, we remark that the volumes of the regions H sE and H TE T (where E and E T are arbitrary) and the region which is their Cartesian product are obviously related by
(A7.2) Appendix 7.1 The canonical distribution for equilibrium states
If instead we take just the volume of the region where H s + H TE, we must choose different limits of integration to obtain:
(A7.3) Appendix 7.1 The canonical distribution for equilibrium states
If we now differentiate this formula with respect to E, we obtain
(A7.4) Appendix 7.1 The canonical distribution for equilibrium states
(p.284) since V(0) = 0.

We can now obtain P s, the distribution function of the system by simply integrating (A7.1) with respect to the variables describing the thermostat, to obtain:

(A7.5) Appendix 7.1 The canonical distribution for equilibrium states
or, using (A7.4) to express A(E):
(A7.6) Appendix 7.1 The canonical distribution for equilibrium states
Let us now remark that if the thermostat is a perfect gas made up of N molecules with mass m, A T is simply the product of V N (where V is the volume occupied by the gas) by the area of the surface of a hypersphere in the 3N-dimensional velocity space, because the equation H T = E may be written in the form Appendix 7.1 The canonical distribution for equilibrium states, and hence A T = V N ω3N(2E/m)(3N−1)/2, where ω3N is the area of the unit sphere in 3N-dimensional space. Then:
(A7.7) Appendix 7.1 The canonical distribution for equilibrium states
and assuming that the energy per particle E/N = 3/(2β) in the thermostat is independent of N (i.e. of the size of the thermostat), as is actually the case for a perfect gas, one obtains:
(A7.8) Appendix 7.1 The canonical distribution for equilibrium states
Let us multiply and divide by A T(E) in (A7.6) and pass to the limit when N → ∞ with the help of (A7.8) (used twice). We finally obtain the canonical distribution:
(A7.9) Appendix 7.1 The canonical distribution for equilibrium states
Note that if we have several systems coupled with a single thermostat, we obtain a canonical distribution with the same value of β for all of them; then this parameter has the properties required from a universal function of temperature. Henceforth we shall no longer need the thermostat and shall therefore suppress the index S in (A7.9). We rewrite it in the following form:
(A7.10) Appendix 7.1 The canonical distribution for equilibrium states
where Z, the partition function, is none other than the integral of e−βH over the entire phase space of the system. The average energy of the system is given by
(A7.11) Appendix 7.1 The canonical distribution for equilibrium states
If we then give a small variation to the parameter β and to one of the other parameters a k (such as volume), we obtain:
(A7.12) Appendix 7.1 The canonical distribution for equilibrium states
We must now introduce the work done on the system by the external forces; the simplest case is obtained by varying a parameter a k in the external forces (and, at least formally, one can always consider this case). Then the work δ* L done on the gas is given by
(A7.13) Appendix 7.1 The canonical distribution for equilibrium states
(p.285) The difference δĒ − δ* L is none other than the heat δ* Q supplied by the system, and hence
(A7.14) Appendix 7.1 The canonical distribution for equilibrium states
It is now clear that the last expression equals Appendix 7.1 The canonical distribution for equilibrium states and is therefore an exact differential. Thus β can be identified with (kT) −1, where k is a constant (the so-called Boltzmann constant) and T the absolute temperature, whereas Appendix 7.1 The canonical distribution for equilibrium states is to be identified (apart from an additive constant) with S/k, where S is the entropy of the system. By a simple calculation we obtain the formula:
(A7.15) Appendix 7.1 The canonical distribution for equilibrium states
From this identification all the formulae of thermodynamics follow. As an example, we consider a perfect gas, for which Z equals V N (2πkT)3N/2 and the gas pressure is given by
(A7.16) Appendix 7.1 The canonical distribution for equilibrium states
i.e. the equation of state of perfect gases with the identification of the relation between the constant k and the gas constant R.

Alternatively, the canonical distribution can be characterized as the one that minimizes the average value of log P under suitable constraints. Gibbs’s considerations on the extretnizing properties start from the elementary inequality:

(A7.17) Appendix 7.1 The canonical distribution for equilibrium states
holding for any real positive x (x may even be zero, provided that we define, by continuity, x logx = 0 for x = 0). The equality sign holds if and only if x = 1. As we remarked in Chapter 6, this reduces, on changing x to 1/x, to the simpler inequality logxx − 1, stating that the logarithmic curve is always below its tangent at (1, 0).

Let us put x = P/G in (A7.17), where P and G are two probability densities in phase space. We obtain the so-called Gibbs inequality:

(A7.18) Appendix 7.1 The canonical distribution for equilibrium states
and the equality sign holds if and only if P and G coincide almost everywhere.

This inequality can be used to prove the minimum properties of the microcanonical and canonical distributions. Note that if we impose the constraint of given total energy and we do not impose any additional constraint besides normalization, then, rather than integrating over the entire phase space, we integrate only over the hypersurface of constant energy and we can let G be a constant in the equation corresponding to (A7.18) and obtain that this constant distribution minimizes the average value of log P (microcanonical distribution). If on the contrary we just assign the average value of energy, letting G = Z −1e−βH in (A7.18), we obtain

(A7.19) Appendix 7.1 The canonical distribution for equilibrium states
(p.286) Since
(A7.20) Appendix 7.1 The canonical distribution for equilibrium states
eqn (A7.19) shows that the canonical distribution minimizes the average value of log P under the constraints of normalization and given average energy.

The Gibbs inequality has several further applications. If for example we assign the one particle distribution function, P (1), and we let Appendix 7.1 The canonical distribution for equilibrium states in (A7.18), we find that the factorized distribution minimizes the mean value of log P and, as an additional result, that the mean value of the logarithm of the one-particle distribution function is not greater than 1/N times the mean value of log P. Another application is provided by the procedure of smoothing P by averaging over small cells, as indicated in the main text of this chapter. In fact, if G is the distribution which is constant over each cell, obtained by averaging P, then it will satisfy (A7.18). If at time t = 0, P is constant over each cell, it will no longer be so at later times, but thanks to the properties of the Lionville equation the average of log P will equal its initial value and hence the initial value of the average of the logarithm of G: the latter average will be at any time no greater than its initial value.