Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

A Functional Calculus Approach to the Ursell-Mayer Functions

C. Y. Shen

Published May 1, 1972• 7 citations
Publisher

Abstract

A more complete treatment of the Ursell-Mayer functions is given here by way of a functional calculus defined on the algebra of complex-valued functions on the exponential of a set. Through measure-theoretical considerations, we are able to establish relationships between these functions and certain linear operators which relate probability densities and correlation functions in statistical mechanics. Also we point out that these functions provide a way of representing states of an infinite system.

DOI
10.1063/1.1666048

Related papers

Dynamics of Ursell — Mayer correlation functions in functional formalism of classical statistics

P. Quaas, K. Voß, P. Ziesche

Acta Physica Academiae Scientiarum Hungaricae•Feb 1, 1968•2 citations

Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. II. Quantum Mechanics in Phase Space

G. S. Agarwal, E. Wolf

In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto $c$-number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Nov 15, 1970•285 citations

Exact Renormalization in the Statistical Mechanics of Fluids

Melville S. Green

Wegner's paradigm for constructing renormalization groups in statistical mechanics is applied to fluids. The coordinates of the fraction ($1\ensuremath{-}y$) of the molecules are integrated over and a system of interactions describing the remaining fraction $y$ is constructed with the aid of the McMillan-Mayer solution theory. Renormalized Ursell functions are defined in terms of which the group operations are shown to be linear. The group operations commute with a scaling operation and it is found that a series of renormalization groups depending on a parameter $x$ can be constructed by combining the operation of integrating over coordinates with the scaling operation. By comparing the renormalized cluster functions with the form of the critical cluster functions given by scaling it is shown that a fixed point of one of the series of groups corresponds to the critical point. The parameter $x$ turns out to be the scaling exponent for the chemical potential. The question of the universality of critical phenomena is expressed as the question of the discreteness or continuity of the spectrum of values of the parameter $x$ for which the linear fixed-point equations have a nontrivial solution. It is suggested that this question can be answered definitively only when a precise formulation is made of the function space in which the sequence of renormalized Ursell functions must lie.

Physical Review A•Oct 1, 1973•7 citations

Functional Integrals and Probabilistic Amplitudes

Laurent Baulieu, J. Iliopoulos, Roland Sénéor

Abstract Functional integrals and probabilistic amplitudes. Brief historical notes. The reconstruction of quantum mechanics from path integrals. The Feynman formulation. Definition and properties of the coherent states and the Bargmann representation.

•Feb 2, 2017•0 citations

The Bogolubov generating functional method in statistical physics and the Wigner transform (Poster Presentation)

Anatoliy K. Prykarpatsky, Nikolai N. Bogolubov

Abstract We show that the N.N. Bogolubov generating functional method is a very effective tool for studying distribution functions of both equilibrium and non equilibrium states of classical many‐particle dynamical systems. In some cases the Bogolubov generating functionals can be represented by means of infinite Ursell‐Mayer diagram expansions, whose convergence holds under some additional constraints on statistical system. The classical Bogolubov idea [1] to use the Wigner density operator transformation for studying the non equilibrium distribution functions is developed and new analytic non‐stationary solution to the classical N.N. Bogolubov evolution functional equation is constructed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

PAMM•Dec 1, 2007•0 citations

On Quantum Statistical Mechanics: A Study Guide

Wladyslaw Adam Majewski

We provide an introduction to a study of applications of noncommutative calculus to quantum statistical physics. Centered on noncommutative calculus, we describe the physical concepts and mathematical structures appearing in the analysis of large quantum systems and their consequences. These include the emergence of algebraic approach and the necessity of employment of infinite-dimensional structures. As an illustration, a quantization of stochastic processes, new formalism for statistical mechanics, quantum field theory, and quantum correlations are discussed.

arXiv:1608.06766•Advances in Mathematical Physics•Jan 1, 2017•22 citations
A more complete treatment of the Ursell-Mayer functions is given here by way of a functional calculus defined on the algebra of complex-valued functions on the exponential of a set. Through measure-theoretical considerations, we are able to establish relationships between these functions and certain linear operators which relate probability densities and correlation functions in statistical mechanics. Also we point out that these functions provide a way of representing states of an infinite system.
In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto ccc-number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.
Wegner's paradigm for constructing renormalization groups in statistical mechanics is applied to fluids. The coordinates of the fraction ($1\ensuremath{-}y$) of the molecules are integrated over and a system of interactions describing the remaining fraction yyy is constructed with the aid of the McMillan-Mayer solution theory. Renormalized Ursell functions are defined in terms of which the group operations are shown to be linear. The group operations commute with a scaling operation and it is found that a series of renormalization groups depending on a parameter xxx can be constructed by combining the operation of integrating over coordinates with the scaling operation. By comparing the renormalized cluster functions with the form of the critical cluster functions given by scaling it is shown that a fixed point of one of the series of groups corresponds to the critical point. The parameter xxx turns out to be the scaling exponent for the chemical potential. The question of the universality of critical phenomena is expressed as the question of the discreteness or continuity of the spectrum of values of the parameter xxx for which the linear fixed-point equations have a nontrivial solution. It is suggested that this question can be answered definitively only when a precise formulation is made of the function space in which the sequence of renormalized Ursell functions must lie.
Abstract We show that the N.N. Bogolubov generating functional method is a very effective tool for studying distribution functions of both equilibrium and non equilibrium states of classical many‐particle dynamical systems. In some cases the Bogolubov generating functionals can be represented by means of infinite Ursell‐Mayer diagram expansions, whose convergence holds under some additional constraints on statistical system. The classical Bogolubov idea [1] to use the Wigner density operator transformation for studying the non equilibrium distribution functions is developed and new analytic non‐stationary solution to the classical N.N. Bogolubov evolution functional equation is constructed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We provide an introduction to a study of applications of noncommutative calculus to quantum statistical physics. Centered on noncommutative calculus, we describe the physical concepts and mathematical structures appearing in the analysis of large quantum systems and their consequences. These include the emergence of algebraic approach and the necessity of employment of infinite-dimensional structures. As an illustration, a quantization of stochastic processes, new formalism for statistical mechanics, quantum field theory, and quantum correlations are discussed.