Publisher Summary

Statistical physics is the study of special laws that govern the behavior and properties of macroscopic bodies. The general character of these laws does not depend on the mechanics that describes the motion of the individual particles in a body, but their substantiation demands a different argument in the two cases. Information concerning the motion of a mechanical system is obtained by constructing and integrating the equations of motion of the system that are equal in number to its degrees of freedom. These statistical laws resulting from the presence of a large number of particles forming the body cannot be reduced to mechanical laws. They cease to have meaning when applied to mechanical systems with a small number of degrees of freedom. Thus, although the motion of systems with a large number of degrees of freedom obeys the same laws of mechanics as that of systems consisting of a small number of particles, the existence of many degrees of freedom results in laws of a different kind.

§ 1 Statistical distributions

Statistical physics, often called for brevity simply statistics, consists in the study of the special laws which govern the behaviour and properties of macroscopic bodies (that is, bodies formed of a very large number of individual particles, such as atoms and molecules). To a considerable extent the general character of these laws does not depend on the mechanics (classical or quantum) which describes the motion of the individual particles in a body, but their substantiation demands a different argument in the two cases. For convenience of exposition we shall begin by assuming that classical mechanics is everywhere valid.

In principle, we can obtain complete information concerning the motion of a mechanical system by constructing and integrating the equations of motion of the system, which are equal in number to its degrees of freedom. But if we are concerned with a system which, though it obeys the laws of classical mechanics, has a very large number of degrees of freedom, the actual application of the methods of mechanics involves the necessity of setting up and solving the same number of differential equations, which in general is impracticable. It should be emphasised that, even if we could integrate these equations in a general form, it would be completely impossible to substitute in the general solution the initial conditions for the velocities and coordinates of all the particles.

At first sight we might conclude from this that, as the number of particles increases, so also must the complexity and intricacy of the properties of the mechanical system, and that no trace of regularity can be found in the behaviour of a macroscopic body. This is not so, however, and we shall see below that, when the number of particles is very large, new types of regularity appear.

These statistical laws resulting from the very presence of a large number of particles forming the body cannot in any way be reduced to purely mechanical laws. One of their distinctive features is that they cease to have meaning when applied to mechanical systems with a small number of degrees of freedom. Thus, although the motion of systems with a very large number of degrees of freedom obeys the same laws of mechanics as that of systems consisting of a small number of particles, the existence of many degrees of freedom results in laws of a different kind.

The importance of statistical physics in many other branches of theoretical physics is due to the fact that in Nature we continually encounter macroscopic bodies whose behaviour can not be fully described by the methods of mechanics alone, for the reasons mentioned above, and which obey statistical laws.

In proceeding to formulate the fundamental problem of classical statistics, we must first of all define the concept of phase space, which will be constantly used hereafter.

Let a given macroscopic mechanical system have s degrees of freedom: that is, let the position of points of the system in space be described by s coordinates, which we denote by qi, the suffix i taking the values 1, 2,…, s. Then the state of the system at a given instant will be defined by the values at that instant of the s coordinates qi and the s corresponding velocities . In statistics it is customary to describe a system by its coordinates and momenta pi, not velocities, since this affords a number of very important advantages. The various states of the system can be represented mathematically by points in phase space (which is, of course, a purely mathematical concept); the coordinates in phase space are the coordinates and momenta of the system considered. Every system has its own phase space, with a number of dimensions equal to twice the number of degrees of freedom. Any point in phase space, corresponding to particular values of the coordinates qi and momenta pi of the system, represents a particular state of the system. The state of the system changes with time, and consequently the point in phase space representing this state (which we shall call simply the phase point of the system) moves along a curve called the phase trajectory.

Let us now consider a macroscopic body or system of bodies, and assume that the system is closed, i.e. does not interact with any other bodies. A part of the system, which is very small compared with the whole system but still macroscopic, may be imagined to be separated from the rest; clearly, when the number of particles in the whole system is sufficiently large, the number in a small part of it may still be very large. Such relatively small but still macroscopic parts will be called subsystems. A subsystem is again a mechanical system, but not a closed one; on the contrary, it interacts in various ways with the other parts of the system. Because of the very large number of degrees of freedom of the other parts, these interactions will be very complex and intricate. Thus the state of the subsystem considered will vary with time in a very complex and intricate manner.

An exact solution for the behaviour of the subsystem can be obtained only by solving the mechanical problem for the entire closed system, i.e. by setting up and solving all the differential equations of motion with given initial conditions, which, as already mentioned, is an impracticable task. Fortunately, it is just this very complicated manner of variation of the state of subsystems which, though rendering the methods of mechanics inapplicable, allows a different approach to the solution of the problem.

A fundamental feature of this approach is the fact that, because of the extreme complexity of the external interactions with the other parts of the system, during a sufficiently long time the subsystem considered will be many times in every possible state. This may be more precisely formulated as follows. Let Δp Δq denote some small “volume” of the phase space of the subsystem, corresponding to coordinates qi and momenta pi lying in short intervals Δqi, and Δpi. We can say that, in a sufficiently long time T, the extremely intricate phase trajectory passes many times through each such volume of phase space. Let Δt be the part of the total time T during which the subsystem was in the given volume of phase space ΔpΔq.† When the total time T increases indefinitely, the ratio Δt/T tends to some limit

(1.1)

This quantity may clearly be regarded as the probability that, if the subsystem is observed at an arbitrary instant, it will be found in the given volume of phase space Δp Δq.

On taking the limit of an infinitesimal phase volume‡

(1.2)

we can define the probability dw of states represented by points in this volume element, i.e. the probability that the coordinates qi and momenta pi have values in given infinitesimal intervals between qi, pi and qi + dqi, pi + dpi. This probability dw may be written

(1.3)

where (p1, …, ps, q1, …, qs) is a function of all the coordinates and momenta; we shall usually write for brevity (p, q) or even simply. The function , which represents the “density” of the probability distribution in phase space, is called the statistical distribution function, or simply the distribution function, for the body concerned. This function must obviously satisfy the normalisation condition

(1.4)

(the integral being taken over all phase space), which simply expresses the fact that the sum of the probabilities of all possible states must be unity.

The following circumstance is extremely important in statistical physics. The statistical distribution of a given subsystem does not depend on the initial state of any other small part of the same system, since over a sufficiently long time the effect of this initial state will be entirely outweighed by the effect of the much larger remaining parts of the system. It is also independent of the initial state of the particular small...