Elementary Principles in Statistical Mechanics

J. Willard Gibbs

• Publisher: Dover Publications
• Year: 2014
• Print ISBN: 9780486789958
• eBook ISBN: 9780486799155
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

Written by J. Willard Gibbs, the most distinguished American mathematical physicist of the nineteenth century, this book was the first to bring together and arrange in logical order the works of Clausius, Maxwell, Boltzmann, and Gibbs himself. The lucid, advanced-level text remains a valuable collection of fundamental equations and principles.
Topics include the general problem and the fundamental equation of statistical mechanics, the canonical distribution of the average energy values in a canonical ensemble of systems, and formulas for evaluating important functions of the energies of a system. Additional discussions cover maximum and minimal properties of distribution in phase, a valuable comparison of statistical mechanics with thermodynamics, and many other subjects.

ELEMENTARY PRINCIPLES IN STATISTICAL MECHANICS

CHAPTER I.

GENERAL NOTIONS. THE PRINCIPLE OF CONSERVATION OF EXTENSION-IN-PHASE.

WE shall use Hamilton’s form of the equations of motion for a system of n degrees of freedom, writing q₁, … q_n for the (generalized) coördinates,

for the (generalized) velocities, and

for the moment of the forces. We shall call the quantities F₁, … F_n the (generalized) forces, and the quantities p₁ … p_n, defined by the equations

where ∊_p denotes the kinetic energy of the system, the (generalized) momenta. The kinetic energy is here regarded as a function of the velocities and coördinates. We shall usually regard it as a function of the momenta and coördinates,* and on this account we denote it by ∊_p. This will not prevent us from occasionally using formulae like (2), where it is sufficiently evident the kinetic energy is regarded as function of the

and q’s. But in expressions like d∊_p/dq₁, where the denominator does not determine the question, the kinetic energy is always to be treated in the differentiation as function of the p’s and q’s.

We have then

These equations will hold for any forces whatever. If the forces are conservative, in other words, if the expression (1) is an exact differential, we may set

where ∊_q is a function of the coördinates which we shall call the potential energy of the system. If we write ∊ for the total energy, we shall have

and equations (3) may be written

The potential energy (∊_q) may depend on other variables beside the coördinates q₁… q_n. We shall often suppose it to depend in part on coördinates of external bodies, which we shall denote by a₁, a₂, etc. We shall then have for the complete value of the differential of the potential energy *

where A₁, A₂, etc., represent forces (in the generalized sense) exerted by the system on external bodies. For the total energy (∊) we shall have

It will be observed that the kinetic energy (∊_p) in the most general case is a quadratic function of the p’s (or

’s) involving also the q’s but not the a’s ; that the potential energy, when it exists, is function of the q’s and a’s ; and that the total energy, when it exists, is function of the p’s (or

’s), the q’s, and the a’s. In expressions like d∊/dq₁ the p’s, and not the

’s, are to be taken as independent variables, as has already been stated with respect to the kinetic energy.

Let us imagine a great number of independent systems, identical in nature, but differing in phase, that is, in their condition with respect to configuration and velocity. The forces are supposed to be determined for every system by the same law, being functions of the coördinates of the system q₁, … q_n, either alone or with the coördinates a₁, a₂, etc. of certain external bodies. It is not necessary that they should be derivable from a force-function. The external coördinates a₁, a₂, etc. may vary with the time, but at any given time have fixed values. In this they differ from the internal coördinates q₁, … q_n, which at the same time have different values in the different systems considered.

Let us especially consider the number of systems which at a given instant fall within specified limits of phase, viz., those for which

the accented letters denoting constants. We shall suppose the differences

, etc. to be infinitesimal, and that the systems are distributed in phase in some continuous manner,* so that the number having phases within the limits specified may be represented by

or more briefly by

where...

Table of contents

• Cover
• Title Page
• Copyright Page
• Preface
• Contents
• Chapter I. General Notions. The Principle of Conservation of Extension–In–Phase.
• Chapter II. Application of the Principle of Conservation of Extension–In–Phase to the Theory of Errors.
• Chapter III. Application of the Principle of Conservation of Extension–In–Phase to the Integration of the Differential Equations of Motion.
• Chapter IV. On the Distribution–In–Phase Called Canonical, in which the Index of Probability is a Linear Function of the Energy.
• Chapter V. Average Values in a Canonical Ensemble Of Systems.
• Chapter VI. Extension–In–Configuration and Extension–In–Velocity.
• Chapter VII. Farther Discussion of Averages in a Canonical Ensemble of Systems.
• Chapter VIII. On Certain Important Functions of the Energies of a System.
• Chapter IX. The Function ϕ and the Canonical Distribution.
• Chapter X. On a Distribution in Phase Called Microcanonical in which all the Systems have the Same Energy.
• Chapter XI. Maximum and Minimum Properties of Various Distributions in Phase.
• Chapter XII. On the Motion of Systems and Ensembles of Systems through Long Periods of Time.
• Chapter XIII. Effect of Various Processes on an Ensemble of Systems.
• Chapter XIV. Discussion of Thermodynamic Analogies.
• Chapter XV. Systems Composed of Molecules.