Entropy

    Let us now turn to the concept of thermal equilibrium. Before we can truly start, we must make some definitions that we will use throughout this course:
Fundamental Assumption of Thermal Physics
A closed system is equally likely to be in any of the quantum states accessible to it.
Closed System
A system that has a constant energy, a constant number of particles, a constant volume and constant values of all external parameters that may influence the system.
Accessible Quantum State
One whose properties are compatible with the physical specifications of the system: the energy of the state must be within the stated range of the energy of the system, and the number of particles must be within the stated range of the number of particles of the system.
    To find the general properties of a system, we take the average of an ensemble. An ensemble is a collection of replicas of the system, one replica for each accessible state. Now, let S denote an accessible state. Then g(s) is the multiplicity function of state s and the probability of the system being in state s is P(s) = l/g(s). From this we get that the ensemble average of an observed quantity X(s) is
 
 
Formula
(3.1)

Now consider two closed systems S1 and S2. Each system has a well defined energy, Ui, and number of particles, Ni. If we bring them into contact so that energy can be freely transferred from one to the other, we call this thermal contact. The two systems can now be viewed as a single larger system, which we denote as S = S1 + S2, with a constant energy U = U1 + U2. We need to know the configuration of the combined system. It is specified by Sl and S2, where we must have S = Sl + S2. The multiplicity of the combined system, S, is
 
 

Formula
(3.2)

We want to find the configuration that maximizes g(N,s). Such a configuration is called the most probable configuration. For large systems we know that the product in (3.2) will have a very sharp maximum. Thus, a relatively small number of configurations will dominate the statistical properties of the combined system.
    Such a sharp maximum is a property of every realistic type of large system for which exact solutions are available. We will postulate that this is a general property of large systems. This allows us to replace the average of a physical quantity over all accessible configurations in (3.2) by an average over only the most probable configuration.

Example:
    Say S1 and S2 are spin systems. Then each system has an energy Ui = -2simB. Assume all the particles have a magnetic moment m. What is the configuration that maximizes the multiplicity?

    Assume that N1 < N2. Then s1 goes from -½N1 to ½N1. For a spin system, g(N,s) is then
 
 

Formula
(3.3)

To find the maximum, we take the derivative of g(N,s) with respect to sl and set it to zero. As before, it is easier to work with logarithms, so we take the logarithm of (3.3) first. Then

Formula




A quick check of the second derivative verifies that this is a maximum. Then the condition for g(N,s) to be maximized is just
 
 

Formula
(3.4)

or in other words, when s1,m and s2,m are the values of s1 and s2 at the maximum. How accurate is this function? Let
 
 

s1 = s1,m + d
 s2 = s2,m - d

Then the number of states for a particular s value is
 
 

Formula




or, since s1/N1 = s2/N2,
 
 

Formula




so, if N1 = N2 = 1022 and d = 1012 (so that d/N1 = 10-10), then
 
 

Formula




which means that the multiplicity function is reduced by a factor of approximately 10-174 from its maximum value.

Thermal Contact

    Now lets generalize this for any two systems in thermal contact. Let the first system, S1, have an energy Ul, and the second system, S2, have an energy U2. Then the total energy, U = U1 + U2, is conserved. In a manner similar to the binary model system, we can write the multiplicity function, g(N,U) as
 
 
Formula
(3.5)

where U1 £ U. As before, we want to find the configuration which maximizes g(N,U). Let an amount of energy dUl be transferred from S1 to S2. Then by conservation of energy we get
 
 

dUl + dU2 = 0



which implies that
 
 

dU1 = -dU2



and, by the chain rule of calculus,
 
 

Formula



The maximum is given by
 
 

Formula



or, dividing through by g1g2,
 
 

Formula
(3.6)

Define the quantity s, called the entropy, to be
 
 

s = ln g(N,U)
(3.7)

Then the condition of thermal equilibrium becomes
 
 

Formula
(3.8)

Notice that the definition of entropy shows that the entropy is nothing more than a measurement of the number of quantum states accessible to the system for a specific energy!