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Non-equilibrium statistical mechanics in the general theory of relativity. I. A general formalism
ABSTRACTS
OF
PAPERS
A Class of Exactly Solvable Potentials. GINOCCHIO. Theoretical Division. Alamos, New Mexico 87545.
TO APPEAR
IN FUTURE
i. One-Dimensional
Los Alamos
National
Schrcdinger
Laboratory,
263
ISSUES
University
Equation.
JOSEPH N. of California, Los
The one-dimensional Schrodinger equation is solved for a new class of potentials with varying depths and shapes. The energy eigenvalues are given in algebraic form as a function of the depth and shape of the potential. The eigenfunctions and scattering function are also given in closed form. For certain shapes these potentials resemble the mean field of an atomic nucleus.
Non-Equilibrium
Statistical Mechanics in the General Theory of Relativity. 1. A General Formalism.
WERNER ISRAEL. Department of Physics and Theoretical Physics Institute. University of Alberta, Edmonton, Alberta T6G 251. Canada: AND HENRY E. KANDRUP. Department of Physics, University of California, Santa Barbara, California 93106 and Center for Studies in Statistical Mechanics and Center for Relativity, University of Texas, Austin, Texas 78712. This is the first in a series of papers, the overall objective of which is the formulation of a new covariant approach to non-equilibrium statistical mechanics in classical general relativity. The object here is the development of a tractable theory for self-gravitating systems. It is argued that the “state” of an N-particle system may be characterized by an N-particle distribution function, defined in an 8Ndimensional phase space, which satisfies a collection of N conservation equations. By mapping the true physics onto a fictitious “background” spacetime, which may be chosen to satisfy some “average” field equations, a useful covariant notion of “evolution” in response to a fluctuating “gravitational force” is then obtained. For many cases of practical interest. it may be supposed that (i) these fluctuating forces satisfy linear field equations and (ii) they may be modeled by a direct interaction. In this case, a relativistic projection operator formalism to derive exact closed equations for the evolution of such objects as an appropriately defined reduced one-particle distribution function can be used. By capturing. in a natural way, the notion of a dilute gas, or impulse. approximation, a comparatively simple equation for the one-particle distribution can be derived. If. furthermore, the effects of the fluctuating forces are treated as “localized” in space and time, a tractable kinetic equation which reduces, in the Newtonian limit, to the standard Landau equation is obtained.
OF
PAPERS
A Class of Exactly Solvable Potentials. GINOCCHIO. Theoretical Division. Alamos, New Mexico 87545.
TO APPEAR
IN FUTURE
i. One-Dimensional
Los Alamos
National
Schrcdinger
Laboratory,
263
ISSUES
University
Equation.
JOSEPH N. of California, Los
The one-dimensional Schrodinger equation is solved for a new class of potentials with varying depths and shapes. The energy eigenvalues are given in algebraic form as a function of the depth and shape of the potential. The eigenfunctions and scattering function are also given in closed form. For certain shapes these potentials resemble the mean field of an atomic nucleus.
Non-Equilibrium
Statistical Mechanics in the General Theory of Relativity. 1. A General Formalism.
WERNER ISRAEL. Department of Physics and Theoretical Physics Institute. University of Alberta, Edmonton, Alberta T6G 251. Canada: AND HENRY E. KANDRUP. Department of Physics, University of California, Santa Barbara, California 93106 and Center for Studies in Statistical Mechanics and Center for Relativity, University of Texas, Austin, Texas 78712. This is the first in a series of papers, the overall objective of which is the formulation of a new covariant approach to non-equilibrium statistical mechanics in classical general relativity. The object here is the development of a tractable theory for self-gravitating systems. It is argued that the “state” of an N-particle system may be characterized by an N-particle distribution function, defined in an 8Ndimensional phase space, which satisfies a collection of N conservation equations. By mapping the true physics onto a fictitious “background” spacetime, which may be chosen to satisfy some “average” field equations, a useful covariant notion of “evolution” in response to a fluctuating “gravitational force” is then obtained. For many cases of practical interest. it may be supposed that (i) these fluctuating forces satisfy linear field equations and (ii) they may be modeled by a direct interaction. In this case, a relativistic projection operator formalism to derive exact closed equations for the evolution of such objects as an appropriately defined reduced one-particle distribution function can be used. By capturing. in a natural way, the notion of a dilute gas, or impulse. approximation, a comparatively simple equation for the one-particle distribution can be derived. If. furthermore, the effects of the fluctuating forces are treated as “localized” in space and time, a tractable kinetic equation which reduces, in the Newtonian limit, to the standard Landau equation is obtained.