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Communications in Theoretical Physics

On Theory of the Statistical Generating Functional for the Order Parameter (III) –Effective Action Formalism for the Order Parameter

Guangzhao Zhou, Zhao-Bin Su, Hao Bai-lin, Lu Yu

Published Jul 1, 1982• 3 citations
Publisher

Abstract

A practical scheme is proposed in this paper to determine the quantum statistical properties of the order parameter in the frame work of the closed time-path Green's functions (CTPGF). As a microscopic quantum theory, this formalism is applicable in principle to both equilibrium and nonequilibrium phenomena and is capable of dealing with statistical systems both above and below their phase transition point. This formalism can be used to discuss the statistical properties of simple as well as composite order parameters in either uniform or nonuniform systems. As simple illustrations and check for the theory, the proposed prescription is applied to the contact-interaction model for superconductivity and a unimode laser system coupled with two-energy-level bound electrons.

DOI
10.1088/0253-6102/1/4/389

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The closed time path Green's function (CTPGF) formalism is applied to the critical dynamics. The related results for the CTPGF approach are briefly reviewed. Three different forms of CTPGF's are defined, transformations from one to another form and other useful computation rules are given. The path integral presentation of the generating functional for CTPGF's is used to derive the Ward-Takahashi identities under both linear and nonlinear transformations of field variables. The generalized Langevin equations for the order parameters and conserved variables are derived from the vertex functional on the closed time path. The proper form of the equations for the conserved variables, including automatically the mode coupling terms, is determined according to the Ward-Takahashi identities and the linear response theory. All existing dynamic models are reobtained by assuming the corresponding symmetry properties for the system. The effective action for the order parameters is deduced by averaging over the random external field. The Lagrangian formulation of the statistical field theory is obtained if the random field one-loop approximation and the second-order approximation of order-parameter fluctuations on different time branches are both taken. The various possibilities of improving the current theory of critical dynamics within the framework of CTPGF's are discussed. The problem of renormalization for the finite-temperature field theory is considered. The whole theoretical framework is also applicable to systems near the stationary states far from equilibrium, whenever there exists an analog of the potential function ("free energy").

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Nonequilibrium generating functional and dynamics of coarse-grained variables

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A generating functional formalism is developed to facilitate the derivation of coarse-grained dynamics of macroscopically relevant variables in various types of many-body problems. The relevant variables can be found either by using symmetry analysis of the action or experimental data. The outcome of the coarse-graining process is a set of coupled partial differential equations for the expectation values of the relevant observables. The set of the macroscopically important observables consists typically of conserved variables and order operators, which are related to the broken symmetries of the system. Symmetry breaking (spontaneous or explicit) in the formalism can be dynamically induced by coupling the system to a generalized heat bath.\n\nThe nonequilibrium generating functional formalism is applied to coarse-graining of generalized hydrodynamics of macroscopic quantum phenomena such as Bose fluids and superconductors, and finally classical fluids. 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Similarly, by symmetry arguments, the order parameter of a weakly interacting Bose fluid is identified and its dynamics in the tree level approximation is shown to follow the Gross-Pitaevskii equation. Higher order corrections can be systematically produced.\n\nClassical molecular dynamics and stochastic cellular automata types of higher level descriptions can be mapped into the same formalism using commonly known techniques. It is also shown how to connect the static and dynamic density functionals to phase-field models (models of critical dynamics), which are commonly used in materials science modelling. A specific example is provided for a liquid-gas-solid system. The phase-field modelling is then applied to the derivation of dynamics of the liquid-gas phase boundaries and triple lines for which a new projection operator technique is developed. We explain in detail how to go from microscopics to macroscopics and relate the effective parameters on each level to each other. 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Navier-Stokes equations are obtained with extra force terms arising from wall-fluid interactions and capillary stresses. It is also shown how the hydrodynamics boundary conditions for stress tensor can be derived from the bulk fields alone in the sharp interface limit. Some mechanisms for generation of the no-slip boundary condition are also hypothesized. These results are important because they show how in general boundary conditions and corrections to them can be derived and how the numerical implementation can benefit from such a formulation of hydrodynamics, where all boundary conditions are expressed in terms of bulk fields alone. Again, these results have wider applicability than the setting where they have been derived.\n\nThe advantage of a systematic coarse-graining procedure enables us to clarify the role of the order parameter concept in the phase-field description of one and two component fluids. We also consider different ways of including gravitational effects and apply dimensional analysis to obtain a condition for the onset of capillary turbulence.\n\nFinally, we present some new formalism for stochastic processes, where the relevant field variables are embedded in an environment characterized by quenched disorder. Whenever some lower dimensional (e.g. defect) structures are projected out of the bulk field evolution equations containing either frozen of coarse-graining generated noise fields, a more complicated effective stochastic noise component will appear in the projected equation of motion of the defect. In particular, the effective noise can be long-ranged even if the quenched background disorder is not. Several condensed matter and hydrodynamical examples describing dynamics of interfaces and line-like objects such as polymers and domain wall defects are considered.

•May 14, 2004•1 citation
A practical scheme is proposed in this paper to determine the quantum statistical properties of the order parameter in the frame work of the closed time-path Green's functions (CTPGF). As a microscopic quantum theory, this formalism is applicable in principle to both equilibrium and nonequilibrium phenomena and is capable of dealing with statistical systems both above and below their phase transition point. This formalism can be used to discuss the statistical properties of simple as well as composite order parameters in either uniform or nonuniform systems. As simple illustrations and check for the theory, the proposed prescription is applied to the contact-interaction model for superconductivity and a unimode laser system coupled with two-energy-level bound electrons.
The closed time path Green's function (CTPGF) formalism is applied to the critical dynamics. The related results for the CTPGF approach are briefly reviewed. Three different forms of CTPGF's are defined, transformations from one to another form and other useful computation rules are given. The path integral presentation of the generating functional for CTPGF's is used to derive the Ward-Takahashi identities under both linear and nonlinear transformations of field variables. The generalized Langevin equations for the order parameters and conserved variables are derived from the vertex functional on the closed time path. The proper form of the equations for the conserved variables, including automatically the mode coupling terms, is determined according to the Ward-Takahashi identities and the linear response theory. All existing dynamic models are reobtained by assuming the corresponding symmetry properties for the system. The effective action for the order parameters is deduced by averaging over the random external field. The Lagrangian formulation of the statistical field theory is obtained if the random field one-loop approximation and the second-order approximation of order-parameter fluctuations on different time branches are both taken. The various possibilities of improving the current theory of critical dynamics within the framework of CTPGF's are discussed. The problem of renormalization for the finite-temperature field theory is considered. The whole theoretical framework is also applicable to systems near the stationary states far from equilibrium, whenever there exists an analog of the potential function ("free energy").
This is the first part of our work on time reversal symmetry applied to non-equi-librium statistical stationary states from a unified microscopic quantum statistical point of view. In this paper, a formalism for time reversal symmetry is constructed in the framework of the Closed Time Path Green's Functions (CTPGF), which can be applied both to equilibrium and non-equilibrium stationary states. By using the generating functional technique of the CTPGF, symmetry relations for the statistical Green's functions and vertex runctions are derived for systems invariant under time reversal.
The closed time path Green's function (CTPGF) formalism is applied to the critical dynamics. The related results for the CTPGF approach are briefly reviewed. Three different forms of CTPGF's are defined, transformations from one to another form and other useful computation rules are given. The path integral presentation of the generating functional for CTPGF's is used to derive the Ward-Takahashi identities under both linear and nonlinear transformations of field variables. The generalized Langevin equations for the order parameters and conserved variables are derived from the vertex functional on the closed time path. The proper form of the equations for the conserved variables, including automatically the mode coupling terms, is determined according to the Ward-Takahashi identities and the linear response theory. All existing dynamic models are reobtained by assuming the corresponding symmetry properties for the system. The effective action for the order parameters is deduced by averaging over the random external field. The Lagrangian formulation of the statistical field theory is obtained if the random field one-loop approximation and the second-order approximation of order-parameter fluctuations on different time branches are both taken. The various possibilities of improving the current theory of critical dynamics within the framework of CTPGF's are discussed. The problem of renormalization for the finite-temperature field theory is considered. The whole theoretical framework is also applicable to systems near the stationary states far from equilibrium, whenever there exists an analog of the potential function (free energy).
A theoretical scheme using closed time-path Green's functions is proposed to describe the quantum statistical properties of the order parameter in terms of a generating functional. The dynamic evolution, is generated by a driving source, while the statistical correlation by a fluctuation source. The statistical causality is shown to hold explicitly and to give rise to a number of important consequences. The problem of determining the quantum statistical properties for the order parameter is reduced to finding a solution of the functional equation for it.
This is the first of a series of papers which describe the functional-integral approach to the study of the statistical and kinetic properties of nonequilibrium quantum fields in flat and curved spacetimes. In this paper we treat a system of self-interacting bosons described by \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$ scalar fields in flat space. We adopt the closed-time-path (CTP or ``in-in'') functional formalism and use a two-particle irreducible (2PI) representation for the effective action. These formalisms allow for a full account of the dynamics of quantum fields, and put the correlation functions on an equal footing with the mean fields. By assuming a thermal distribution we recover the real-time finite-temperature theory as a special case. By requiring the CTP effective action to be stationary with respect to variations of the correlation functions we obtain an infinite set of coupled equations which is the quantum-field-theoretical generalization of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Truncation of this series leads to dissipative characteristics in the subsystem. In this context we discuss the nature of dissipation in interacting quantum fields.To one-loop order in a perturbative expansion of the CTP effective action, the 2PI formalism yields results equivalent to the leading 1/N expansion for an O(N)-symmetric scalar field. To higher-loop order we introduce a two-time approximation to separate the quantum-field effects of radiative correction and renormalization from the statistical-kinetic effects of collisions and relaxation. In the weak-coupling quasiuniform limit, the system of nonequilibrium quantum fields can subscribe to a kinetic theory description wherein the propagators are represented in terms of relativistic Wigner distribution functions. From a two-loop calculation we derive the Boltzmann equation for the distribution function and the gap equation for the effective mass of the quasiparticles. One can define an entropy function for the quantum gas of quasiparticles which satisfies the H theorem. We also calculate the limits to the validity of the binary collision approximation from a three-loop analysis. The theoretical framework established here can be generalized to nonconstant background fields and for curved spacetimes.
A generating functional formalism is developed to facilitate the derivation of coarse-grained dynamics of macroscopically relevant variables in various types of many-body problems. The relevant variables can be found either by using symmetry analysis of the action or experimental data. The outcome of the coarse-graining process is a set of coupled partial differential equations for the expectation values of the relevant observables. The set of the macroscopically important observables consists typically of conserved variables and order operators, which are related to the broken symmetries of the system. Symmetry breaking (spontaneous or explicit) in the formalism can be dynamically induced by coupling the system to a generalized heat bath.\n\nThe nonequilibrium generating functional formalism is applied to coarse-graining of generalized hydrodynamics of macroscopic quantum phenomena such as Bose fluids and superconductors, and finally classical fluids. The method is by no means restricted to fluids, but can be applied to condensed matter problems as well. High density systems are in general more problematic and it remains to be seen if one can come up with more powerful approximation methods within the current formalism in the future.\n\nThe generating functional formalism is shown to reproduce in the specific limits the well-known ground state, finite temperature and time-dependent density functional theories in the quantum regime. We enlarge the set of relevant variables to include also non-conserved order-operators and show how to form generalized density functional theories, which can be by construction linked to the phase-transition behaviour of the system. They are extensions of the traditional density functional theories that were not developed to describe phase changes but only facilitated the expression of the system's (free) energy in terms of the conserved density variable (e.g. number density of electrons).\n\nWe develop a generalized density functional theory (effective action of the generating functional) for superconductors and superfluids. For superconductors it is shown by simple symmetry analysis that the relevant order parameter is the pair field of two electrons. Coarse-graining out the phononic degrees of freedom is shown to lead to a retarded effective interaction between the electrons. In the local static limit (effective interaction between electrons static) the theory reduces to the time-dependent Ginzburg-Landau theory. Similarly, by symmetry arguments, the order parameter of a weakly interacting Bose fluid is identified and its dynamics in the tree level approximation is shown to follow the Gross-Pitaevskii equation. Higher order corrections can be systematically produced.\n\nClassical molecular dynamics and stochastic cellular automata types of higher level descriptions can be mapped into the same formalism using commonly known techniques. It is also shown how to connect the static and dynamic density functionals to phase-field models (models of critical dynamics), which are commonly used in materials science modelling. A specific example is provided for a liquid-gas-solid system. The phase-field modelling is then applied to the derivation of dynamics of the liquid-gas phase boundaries and triple lines for which a new projection operator technique is developed. We explain in detail how to go from microscopics to macroscopics and relate the effective parameters on each level to each other. For example, we express the macroscopic surface tensions from the phase-field model in terms of the molecular dynamics simulation parameters. We also discuss how to extract the relevant cross-over time and length scales from the theoretical description and relate them to the experimentally measurable ones. It is shown how the coupling of the triple line and the liquid-gas boundary leads to emergence of memory effects. For confined geometry we also derive a new form for the restoring force acting on the contact line. The phase-field model of the liquid-gas-solid system is first discussed without explicit hydrodynamics from the point of view of diffuse mass transfer and then hydrodynamic effects (momentum conservation) are added.\n\nEvolution equations are derived for the mass, momentum and energy densities representing the proper set of macro variables of simple fluids. The derivation is based on a macroscopic Poisson bracket formulation of phase-field dynamics. Navier-Stokes equations are obtained with extra force terms arising from wall-fluid interactions and capillary stresses. It is also shown how the hydrodynamics boundary conditions for stress tensor can be derived from the bulk fields alone in the sharp interface limit. Some mechanisms for generation of the no-slip boundary condition are also hypothesized. These results are important because they show how in general boundary conditions and corrections to them can be derived and how the numerical implementation can benefit from such a formulation of hydrodynamics, where all boundary conditions are expressed in terms of bulk fields alone. Again, these results have wider applicability than the setting where they have been derived.\n\nThe advantage of a systematic coarse-graining procedure enables us to clarify the role of the order parameter concept in the phase-field description of one and two component fluids. We also consider different ways of including gravitational effects and apply dimensional analysis to obtain a condition for the onset of capillary turbulence.\n\nFinally, we present some new formalism for stochastic processes, where the relevant field variables are embedded in an environment characterized by quenched disorder. Whenever some lower dimensional (e.g. defect) structures are projected out of the bulk field evolution equations containing either frozen of coarse-graining generated noise fields, a more complicated effective stochastic noise component will appear in the projected equation of motion of the defect. In particular, the effective noise can be long-ranged even if the quenched background disorder is not. Several condensed matter and hydrodynamical examples describing dynamics of interfaces and line-like objects such as polymers and domain wall defects are considered.