Invariant Imbedding Equations for the Dissipation Functions of an Inhomogeneous Finite Slab with Anisotropic Scattering
Two complete sets of differential-integral equations are derived for the dissipation functions of an inhomogeneous finite slab which scatters anisotropically. The first set is for the case of monodire...
Two complete sets of differential-integral equations are derived for the dissipation functions of an inhomogeneous finite slab which scatters anisotropically. The first set is for the case of monodire...
Group Formulation of Potential Scattering
Potential scattering is formulated in terms of an integral equation on the group space of O3. The potential need not be spherically symmetric except at large distances from the scattering center.
Potential scattering is formulated in terms of an integral equation on the group space of O3. The potential need not be spherically symmetric except at large distances from the scattering center.
Potential-Correlation Function Duality in Statistical Mechanics
J. Math. Phys. 8, 2143 (1967); doi:10.1063/1.1705132
Issue Date: October 1967
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The Feynman path-integral formulation is used to analyze the grand-partition function Z and the n-body Green's function Gn or rather their cumulants (or connected parts) 
n for a system governed by p-body interaction potentials vp. It is shown that the functional relationship expressing n in terms of vp is invariant under the transformation exchanging −(−)mm and vm everywhere. Under the same transformation log Z undergoes a change of sign. The content of these results is discussed in conclusion.
©1968 The American Institute of Physics
| History: | Received 12 April 1967 |
| Permalink: |
http://link.aip.org/link/?JMAPAQ/8/2143/1 |
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REFERENCES (10)
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- J. Yvon, Act. Sci. Ind. 203 (1935), to quote the earliest attempt known to the authors, where the one-body potential is eliminated in favor of the “observable” one-body density, yielding, in particular, the so-called virial expansion.
- See, for example, the review article of C. Bloch, Studies in Statistical Mechanics (North-Holland Publishing Company, Amsterdam, 1965), Vol. 3, p. 1.
- A perturbation expansion approach to this result viewed in the framework of field theory together with comments on possible applications to “bootstrap” theory of interaction is being published elsewhere; F. Englert and C. De Dominicis, Nuovo Cimento (to be published).
- Section 20 of Ref. 2 discusses the pseudoperiodicity conditions imposed on many-“time” potentials.
- See, for example, N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Interscience Publishers, Inc., New York, 1959),
- and J. S. Bell in The Many-Body Problem, E. Caianiello, Ed. (Academic Press, Inc., New York, 1962), for its quantum statistical mechanics aspect.
- Results of Table I remain valid if the factors (i)p+q or (i)n+m are suppressed everywhere (this immediately follows from the structure of the diagram expansions, for example). In that form, however, the propagator has an opposite sign. Results of Ref. 3 for field theory are recovered by the substitutions
and
, where
is the usual energy variable. - See, for example, S. F. Edwards,
Phil. Mag. 4, 1171 (1959) for classical systems; - B. Mühlschlegel and H. Zittartz,
Z. Physik 175, 553 (1963) for Ising systems. - J. Yvon, Les corrélations et I'entropie en mécanique statistique classique (Cie Dunod, Paris, 1966). Equations (19) and (20) of the text are his equations (4) and (5), Chap. 2, Sec. 2.
- J. Yvon, Cours de Mécanique Statistique à la Faculté des Sciences de Paris (1966).
- C. De Dominicis and F. Englert (to be published).
