Statistical Physics, Part 1

Author: L.D.Landau et al.
Publisher:
Publish Date: 1999-05-01
Features: In this edition the book has been considerably augmented and revised, with the assistance of L.P. Potaevskil throughout. New sections have been added on the magnetic properties of gases, the thermodynamics of degenerate plasma, liquid crystals, the fluctuation theory of phase transitions of the second kind, and critical phenomena. This book is in English.
Excerpt: 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 cannot 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 coordinates, which we denote by q, 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 coordinates q, and the corresponding velocities q,. In statistics it is customary to describe a system by its coordinates and momenta p, 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 q, and momenta p 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 simply call 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, then 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 Aq, Ap denote some small "volume" of the phase space of the subsystem, corresponding to coordinates q, and momenta p applying in short intervals Aq, and Ap. 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 b be the part of the total time T during which the subsystem was in the given volume of phase space Aq, Ap? When the total time T increases indefinitely, the ratio b/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 Aq, Ap. On taking the limit of an infinitesimal phase volume dq dp = dq1 dq2 ... dq, dp1 dp2 ... dp, (1.2) we can define the probability dw of states represented by points in this volume element, i.e., the probability that the coordinates q and momenta p have values in given infinitesimal intervals between q1, p1 and qd, pd. This probability dw may be written where (p, ..., p1, q, ..., qd) is a function of all the coordinates and momenta; we shall usually write for brevity (p, q) or even simply. The function g, which represents the "density" of the probability distribution in phase space, is called the statistical distribution function, or simply the. For brevity, we shall usually say, as is customary, that the system "is in the volume pq of phase space", meaning that the system is in states represented by phase points in that volume. In what follows we shall always use the conventional notation dp and dq to denote the products of the differentials of all the momenta and all the coordinates of the system respectively.