Statistical physics (2005)

tally negligible e ect of δE in practical cases; iii) the e ective proportionality of ln W (E), and thus of S, to N .

In the general case, the particles may have other degrees of freedom (in the example of diatomic molecules : rotation, vibration), they may be subjected to a potential energy, etc. Results i), ii), iii) are still valid.

1.4General Properties of the Statistical Entropy

As just explained on a special case, the physical parameter that contains the statistical character of the system under study is the statistical entropy. We now define it on more general grounds and relate it to the information on the system.

1.4.1The Boltzmann Definition

The expression (1.16) of S given by Boltzmann relies on the postulate of equal probability of the W accessible microstates of an isolated system. S increases as the logarithm of the microstates number.

1.4.2The Gibbs Definition

A more general definition of the statistical entropy was proposed by Gibbs :

This definition applies to an assembly of systems constructed under the same macroscopic conditions, made up in a similar way according to the process defined in §1.3.2; in the ensemble of all the reproduced systems, pi is the probability to attain the particular state i of the system, that is, the microstate i. It is this Gibbs definition of entropy that will be used in the remaining part of this course. For example, for systems having reached thermal equilibrium at temperature T , we will see that the probability of realizing a microstate i is higher, the lower the microstate energy (pi is proportional to the Boltzmann factor, introduced in §2.4.1). The Boltzmann expression (1.16) of the entropy is a special case, which corresponds to the situation where all W microstates are equiprobable, so that the probability of realizing each of them is pi = 1/W .

16 Chapter 1. Statistical Description of Large Systems

i

It is not possible to proceed the other way and to deduce the general Gibbs definition from the particular Boltzmann definition. Yet let us try to relate the Gibbs and the Boltzmann definitions using a particular example : we are considering a system made up of n1 + n2 equiprobable microstates regrouped into two subsystems 1 and 2. Subsystem 1 contains n1 microstates, its probability of realization is p1 = n1/(n1 + n2). In the same way, subsystem 2 contains n2 microstates, its probability of realization is equal to p2 = n2/(n1 + n2).

From the Boltzmann formulation, the statistical entropy of the total system is

This entropy is related to the absence of information on the microstate of the total system in fact realized :

–we do not know whether the considered microstate belongs subsystem 1 or 2 : this corresponds to the entropy S12, which is the Gibbs term that we want to find in this example;

–a part S1 of the entropy expresses the uncertainty inside system 1 owing to its division into n1 microstates

and is weighted by the probability p1.

In the same way S2 comes from the uncertainty inside system 2

Since the various probabilities of realization are multiplicative, the corresponding entropies, which are related to the probabilities through logarithms, are additive :

isolated system of energy between E and E + δE, where δE is the uncertainty, all the W (E) accessible microstates are equally probable. The statistical entropy S is then defined by

S = kB ln W (E)

where kB = R/N is the Boltzmann constant, with R the ideal gas constant and N the Avogadro number.

The general definition of the statistical entropy, valid for an ensemble average, is

S = −kB pi ln pi

i

where pi is the probability of occurrence of the particular microstate i of the system. The expression for an isolated system is a special case of the latter definition. The statistical entropy is an extensive parameter, which is equal to zero when the microscopic state of the system is perfectly known.

The obtained determinant is developed to first order in dt :

Consequently, the volume around a point of the 6N -dimensional phase space is conserved during the time evolution of this point; the number of systems in this volume, i.e., D(. . . qi, . . . , . . . pi, . . . , t)dτ , is also conserved.

Each system evolves over time, so that it may leave the considered volume while other systems may enter it.

p1

q˙1(t)dt

dq1

dp1

(q˙1(t) + ∂q˙1(t) dq1) dt ∂q1

Fig. 1.6: In the volume dq1 · dp1 (thick line), between the times t and t + dt, the systems which enter were in the light grey rectangle at t ; those which leave the volume were in the dark grey hatched rectangle. It is assumed that q˙1(t) is directed toward the right.

Takea “box”, like the one in Fig. 1.6 which represents the situation for the only coordinates q1, p1. If only this couple of conjugated variables is considered, the

problem involves a single coordinate q1 in real space (so that the velocity q˙1 is necessarily normal to the “faces” at constant q1) and we determine the variation of the number of systems on the segment for the q1 coordinate

[q1, q1 + dq1].

Considering this space coordinate, between the times t and t + dt, one calculates the di erence between the number of systems which enter the volume through the faces normal to dq1 and the number which leave through the same faces : the ones which enter were at time t at a distance of this face smaller or equal to q˙1(t)dt (light grey area on the Fig.), their density is D(q1 −dq1, . . . qi, . . . q3N , p1, . . . pi, . . . p3N ). Those which leave were inside the

The resulting change, between t and t + dt and for this set of faces, of the number of particles inside the considered volume, is equal to

For the couple of “faces” at constant p1, an analogous evaluation is performed, also contributing to the variation of the number of particles in the considered “box.” Now at 3N dimensions, for all the “faces,” the net increase of the number of particles in dτ during the time interval dt is

This is related to a change in the particle density in the considered volume through to ∂D∂t dtdτ . Thus the net rate of change is given by

From the Hamilton equations, the factor of D is zero and the above equation is equivalent to

where dD(q1, . . . , qi, . . . q3N , p1, . . . , pi, . . . p3N ) is the time derivative of the dt

probability density when one moves with the representative point of the system in the 6N -dimensional phase space. This result is equivalent to the one of §A.