


Question
What is temperature? Is temperature a measure of the vibrations of matter? If so isn't it related to kinetic energy and therefore not a base quantity?
Asked by: Neale Denton
Answer
Temperature is really a quantity that emerges in the statistical description of many
particle systems. At a fixed value for the temperature and the pressure of a system, the
internal energy is not fixed, but can fluctuate around a mean value. If the system
consists of a large number of particles then these fluctuations can be ignored.
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 EE1. The number of states available for the heath bath is thus
W(EE1) 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

