Dec., 1962
kh’ATISTICAL
MECHANICS WITH
INTXGHALS O P H E H THAN E N E H G Y
1039
STATISTICAL MECHANICS OF DYNAMICAL SYSTEMS WITH INTEGRALS OTHER THAN ENERGY BY HAROLD GRAD Inslitule for Mathematics and Mechnics, New York University, New York 3, N . Y . RdCdVsd AUQU64 16, 1068
Statistical mechanics is generalized to include a complete set of integrals. This leads to a generalization of classical thermodynamics and generalized macroscopic conservation equations. Application is made to the integral of angular momentum and to thc theory of irreversible thermodynamics.
Introduction C:onveiitionally, statistical mechanics atid thermodynamics are based on the energy, fluid dynamics on momentum as well, while the other integrals of classical particle dynamics, notably the angular momentum, are ignored or relegated to a footnote. If the theory is generalized to include a complete set of integrals, it is found possible to give a more rational presentation to the whole of classical statistical mechanics. The aim of stat,istical mechanics is the derivation of the macroscopic properties of matter entirely from the microscopic properties of molecules. It is found necessary to supplement molecular information by two fundamental postulates: (1) the number of time-independent integrals is small compared to the total number of degrees of freedom-it is usually assumed, implicitly, that there is only one integral, the energy; (2) the probability distribution in phase space is absolutely continuous-this supplants the more restrictive hypothesis of equal a priori probabilities to equal volumes. The first result is the derivation of a generalization of the microcanonical distribution for an isolated system. Then follow derivations of a generalized canonical distribution for a system in “thermal” equilibrium with its surroundings, grand canonical distribution for a system free to exchange molecules with its surroundings, and a further distribution valid for a system in mechanical equilibrium with the surroundings. Corresponding to each of these physical situations is its own thermodynamical structure with a few significant differences. In particular, the entropy function is different in the canonical and grand canonical cases, thereby resolving the Gibbs Paradox purely classically, without recourse to quantum mechanics. Corresponding to the set of integrals is a set of generalized temperatures which, in “thermal” equilibrium, are equal to those of the surroundings. For the grand canonical distribution this is true of the chemical potentials and in mechanical equilibrium of certain general “forces.” A generalized expression of the second law is discovered in which appears an entropy with the usual “increasing” properties. In non-equilibrium, conservation equations are derived for each integral, and generalizatioiis of the usual results of irreversible thermodynamics are obtained by consideration of the entropy production. In particular, interesting results are obtained for the integral of angular momentum. In a rigid body rotation, additional “hidden” internal energy and angular momentum are found; equiparti-
tioii of energy is lost; chemical equilibria are dcpendent on angular velocity. In noli-ecliiilihrium, the stress tensor is found to be asymmetric in general (this effect is very small, however), and thc force system in a fluid is generalized to include a couple per unit area as well as a force per unit aroa. This talk is based on a paper‘ to which references are made for proofs and further discussion of some results; the material on the grand canonical distribution and the derivation of the conservation equations supplements that paper. Statistical Mechanics and Thermodynamics
We assume the existence of a time-independent Hamiltonian H(qi, pi), i = 1 . . . s, and the validity of the canonical equations
Solution of these equations gives the motion of the system, qi(t), p i ( t ) ; geometrically this is a curve with parameter t (path) in the 2s-dimensional phase spuce P:(qi, pi). In this way the differential equations (1) define a steady flow of the phase space into itself with parameter t. This flow is incompressible, and the volume element d P is preserved since the divergence of the flow velocity vanishes
A time-independent integral is a function e ( P ) which has a constant value on each path. Alternatively, every path must lie entirely within a (2s ])-dimensional manifold E ( P )= constant. If we haverdistinct integrals, Q ( P ). . . a,(!‘), each path will lie in a (2s - r)-dimensional intersection of the manifolds ei(P) = constant, and, if T = 2s - 1, thc intersection is the path itself. In general there (lo not exist as many as 2s - 1 independent integrals, and in this case we may call the system ergodic. For example, on a billiard table the phase space is ) velocities four-dimensional, coordinat,es ( 2 , ~ and (u,v). Two integrals are u2and v 2 since either u or v changes sign on reflection at a cushion; however, no third integral exists which is not a function of u2 and v2. The reason for this is that all paths (except the periodic ones-of measure zero) meander about the whole table coming arbitrarily close to every point (qy) and even spending, on the avcrage, equal times in equal areas. This is the cme
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(1) H.
Grad, Con” Purc and A p p l . Math., 6 , KO.4 (1932).
HAROLD GRAD
1040
more generally for any ergodic system; almost every path "fills" the (2s r)-manifold to which it belongs and, in particular, the infinite time average of any phase function @ ( P )over a path is equal to the corresponding (29 - r)-dimensional phase average. Our fundamental postulate is that the number of integrals, P, is small compared to 2s. This is, in a sense, a gap in the theory since this postulate is, in principle, provable for any given dynamical system. However t,here exist mathematical theorems which (interpreted broadly) assert that almost any system chosen at random will have no integrals other than the known Such a statement is, for our purposes, even more significant than would be the proof of ergodicity of any particular dynamical system. Following the usual procedure when it is impossible practically to obtain complete information about the state of a system ( i e , , all the coordinates qi and p i ) , we introduce an a priori robability density f(P) with the property that f(P)dP is the probability of finding the state P within the domain D . Implied is the possibility of generating the probability f(P) by repeated experiments with the macroscopic variables fixed. This is the second postulate, namely, that no finite probability is concentrated on a manifold of dimension lower than the full phase space; the cumulative probability distribution is absolutely continuous. The state of a system is now specified by a function f ( P ) instead of by a point P. In general, the probability will vary with time, f(P,t). The variation in time is determined by the flow in the phase space, explicitly by Liouville's equation
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L
Vol. 56
It is convenient to use the uniform notation xi, i = 1 . . . 29 for (pi, p i ) , i = 1 . . . s; also we write z
and e for Xi and ei and dz and de for the elements of volume in 2s and in r-space. The bounded domain V , defined by ei < Ci has volume V(c) (3)
and we define (4)
Q(c) is the limit of the volume of a thin shell ci < ti < ci dci divided by dc. It is easily proved that
+
Jj(r)dz
= Jbf(e)n(.)d.
(5)
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