Asked by: Neale Denton

Suppose you are given an isolated system with a very large number of particles, e.g. a gas or a solid. If all you know of this system is that its internal energy is E and its volume is V, then clearly there are many, many different states the system could be in. Let's denote the number of states compatible with internal energy E and volume V as W(E,V). It can be shown that all these states are equally likely. To define temperature consider what happens if you bring two systems in contact with each other. If system 1 has energy E1 and system 2 has energy E2 then, then for each of the W1(E1) states available for system 1 there are W2(E2) states for system 2, so the total number of states for the combined system is the product W1(E1)* W2(E2). Let's denote the total energy as E = E1+ E2. If the total energy of the whole system is fixed then there will be a particular value for E1 and E2 such that the product W1(E1)* W2(E2) is maximal. Since all states are equally likely, this is the most likely situation. If you bring systems 1 and 2 into contact, and E1 and E2 are chosen arbitrary, then heat will flow from one system into the other until W1(E1)* W2(E2) is maximal. This situation thus corresponds with thermal equilibrium. If W1(E1)* W2(E2) is maximal, then clearly Log[W1(E1)* W2(E2)] is also maximal. Equating the derivative w.r.t. E1 to zero, yields:

d Log (W1)/dE1 + d Log (W2)/dE1 = 0 (1)

Since E2 = E - E1, the derivative of W2 w.r.t E1 can be replaced by minus the derivative w.r.t. E2. This gives:

d Log (W1)/dE1 = d Log (W2)/dE2 (2)

The temperature T of a system is defined as:

1/(kT) = d Log (W)/dE (3)

Here k is Boltzmann's constant.

Equation (2) then implies that temperatures are equal in thermal equilibrium.

Knowing the temperature of a system allows you to calculate the probability that the system is in a particular state. If you consider the system in contact with a heath bath at temperature T as an isolated system then all states are equally likely for the combined system. If the combined system has energy E, and the subsystem is in one particular state with energy E1 then the heath bath has to have energy E-E1. The number of states available for the heath bath is thus W(E-E1) and this is the total number of states of the whole system because the subsystem is in one particular state. The probability P for the subsystem to be in a particular state with energy E1 is thus given as:

Log (P) = Log(W(E- E1)) = Log(W(E)) - E1* d Log (W)/dE + ... (4)

Using (3) we can write this as

P = Exp[-E1/kT]/Z (5)

With Z a normalization factor.

The higher order terms are absent because they are proportional to the change in temperature of the heath bath as a function of E1, but in the case of an ideal heath bath these terms are zero.

Using (5) it is in principle possible to derive the precise relation between internal energy and temperature. The precise relation, however, depends on the system in question.

Answered by: Saibal Mitra, M.S., Physics Grad Student, uva Amsterdam

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