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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Narrow resonances in statistical mechanics

Roger F. Dashen, RAMAMURTI RAJARAMAN

Published Jul 15, 1974• 64 citations
Publisher

Abstract

It is shown that when the dynamics of a system is dominated by narrow-resonance formation it behaves like a noninteracting system with added species of free particles, corresponding to all the quantum numbers and the statistics of each of the resonances. This equivalence, widely assumed and used in practice, is established by explicitly constructing many-particle $S$ matrices consistent with unitarity that correspond to purely resonance-dominated dynamics. These are then used in the $S$-matrix formulation of statistical mechanics to obtain the grand partition function, which is seen to reveal the above result. In the following paper, operational criteria are presented for determining when in general narrow resonances and loosely bound states may be treated as "elementary" particles.

DOI
10.1103/physrevd.10.694

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It is shown that when the dynamics of a system is dominated by narrow-resonance formation it behaves like a noninteracting system with added species of free particles, corresponding to all the quantum numbers and the statistics of each of the resonances. This equivalence, widely assumed and used in practice, is established by explicitly constructing many-particle SSS matrices consistent with unitarity that correspond to purely resonance-dominated dynamics. These are then used in the SSS-matrix formulation of statistical mechanics to obtain the grand partition function, which is seen to reveal the above result. In the following paper, operational criteria are presented for determining when in general narrow resonances and loosely bound states may be treated as "elementary" particles.
We have formulated the statistical mechanics in terms of the SSS matrix, which describes the scattering processes taking place in the thermodynamical system of interest. Such a formulation is necessary for studying the systems whose microscopic constituents behave according to the laws of relativistic quantum mechanics. Our result is a simple prescription for calculating the grand canonical potential of any gaseous system given the free-particle energies and SSS-matrix elements. When applied to a nonrelativistic gas, it gives a simple prescription for calculating all virial coefficients. Simplified relativistic gas models are considered as examples of application. A general form of the Levinson's Theorem for any number of particles follows immediately from our formalism. Its applications in statistical mechanics are briefly discussed.
We have formulated the statistical mechanics in terms of the SSS matrix, which describes the scattering processes taking place in the thermodynamical system of interest. Such a formulation is necessary for studying the systems whose microscopic constituents behave according to the laws of relativistic quantum mechanics. Our result is a simple prescription for calculating the grand canonical potential of any gaseous system given the free-particle energies and SSS-matrix elements. When applied to a nonrelativistic gas, it gives a simple prescription for calculating all virial coefficients. Simplified relativistic gas models are considered as examples of application. A general form of the Levinson's Theorem for any number of particles follows immediately from our formalism. Its applications in statistical mechanics are briefly discussed.
The fundamentals of Statistical Mechanics require a fresh definition in the context of the developments in Classical Mechanics of integrable and chaotic systems. This is done with the introduction of Micro Partitions ; a union of disjoint components in phase space. Partition functions including the invariants, Kolmogorov Entropy and Euler number are introduced. The ergodic hypothesis for partial ergodicity is discussed. In the context of Quantum Mechanics the presence of symmetry groups with irreducible representations gives rise to degenerate and non degenerate spectrum for the Hamiltonian. Quantum Statistical Mechanics is formulated including these two cases ; by including the multiplicity dimension of the group representation and the Casimir invariants into the Partition function. The possibility of new kinds of phase transitions is discussed. The occurence of systems with non simply connected configuration spaces and Quantum Mechanics for them, also requires a possible generalisation of Statistical Mechanics. The developments of Quantum pure, mixed, and entangled states has made it neccessary to understand the Statistical Mechanics of the multipartite N particle system. And to obtain in terms of the density matrices, written in energy basis, the Trace of the Gibbs operator as the Partition function and use it to get statistical averages of operators. There are some issues of definition, interpretation and application that are discussed.
We establish a rather deep relation between statistical mechanics and quantum mechanics at an imaginary time. For the quantum field theory, this implies a relation between finite temperature quantum field theory in Minkowski space and quantum field theory in Euclidean space with one compact Euclidean time direction. The quantum mechanical language being a proper formulation of statistical physics as a molecular theory is basically the study of quantum equilibrium as well as non-equilibrium systems. A quantum system found in a thermodynamic equilibrium state with an external classical system (thermostat or thermal reservoir) may be characterized by fixed values of thermodynamic parameters of the system, say, for example, temperature T , volume V and number of particles N . In this way the total energy of the thermostat is assumed to be much greater than the energy of the system. The behavior of the quantum system may be assumed to be independent of the nature of the thermostat. The most convenient object of the quantum theory to describe such non-isolated systems is the density matrix since the partition function does not determine any local thermodynamic quantities. It is for this reason that we introduced quantum mechanical and quantum statistical density matrices which are necessary to do the investigation via path integration. This enables us to make a transition to the quasi-classical approximation of quantum statistical mechanics, which, in turn, enables us to evaluate the first quantum correction to the classical Boltzmann distribution function.