Equilibrium Correlation Functions Research Articles

Correlation functions for dissipative two-state systems: Effects of the initial preparation.

We study the influence of system-bath correlations in the initial state on the dynamics of biased two-state systems. The formal solution for the equilibrium correlation function in the form of a power series in the number of transitions between the system states is determined. For Ohmic damping and a special value of the coupling strength (K=1/2), the exact solution for all times and temperatures is presented. We find that at low temperatures the long-time tails of the decay into equilibrium are strongly affected by the initial preparation. Correspondingly, the low-frequency behavior of the neutron-scattering function is qualitatively changed by the presence or absence of correlations in the initial state.

Nonlinear optical response in condensed phases: A microscopic theory using the multipolar Hamiltonian.

A general scheme is presented for calculating the nonlinear optical response in condensed phases that provides a unified picture of excitons, polaritons, retardation, and local-field effects in crystals and in disordered systems. A fully microscopic starting point is taken by considering the evolution of the quantized radiation-matter system described by the multipolar (\ensuremath{\mu}\ensuremath{\cdot}${\mathbf{D}}^{\mathrm{\ensuremath{\perp}}}$) Hamiltonian. For a molecular system with localized electronic states we derive equations of motion in which the instantaneous intermolecular interactions are explicitly recovered and the interaction with the Maxwell electric field is of the \ensuremath{\mu}\ensuremath{\cdot}${\mathbf{E}}^{\mathrm{\ensuremath{\perp}}}$ type. It is shown that with total neglect of retardation, these equations lead to the usual expressions for nonlinear optical susceptibilities in terms of equilibrium correlation functions of the polarization field of the molecular system. A mean-field approximation for these equations of motion yields the commonly used local-field expression. Finally, a procedure is proposed for calculating the optical response that fully accounts for retardation, based on a hierarchy of equations of motion for polaritons.

Effects of dynamic screening on the electrical conductivity of fully ionized, nondegenerate hydrogen plasma

The electrical conductivity of fully ionized, nondegenerate hydrogen plasma is expressed within the Zubarev method by equilibrium correlation functions. Using the Green’s function technique, the Lenard–Balescu–Gurnsey collision integral of a generalized Boltzmann equation is derived that accounts for the effects of dynamic screening. Applying the usual random phase approximation, numerical results for the collision integral and the electrical conductivity are compared with the case of static screening (ω=0) and the long-wavelength limit (q→0) for the dielectric function Ε(q,ω). Effective low-density expansions are given for the collision integrals as well as for the electrical conductivity that are applicable for a wide range of density and temperature.

Spatial effects on the fluctuations of a nuclear power reactor

The effects of spatial inhomogeneities in a nuclear system are studied by using the compounding moments method. In particular, the neutron density and temperature equilibrium correlation functions are explicitly calculated for a realistic linearized nuclear reactor model described in terms of a master equation.

Relaxation of the Spin Autocorrelation Function in the Kinetic Ising Model with Bond Dilution

The relaxation of the equilibrium correlation function q ( t )= N -1 Σ i =1 N is studied by the Monte Carlo method for the bond-diluted kinetic Ising model on the square lattice with a bond concentration below the percolation threshold. Here, the system has N Ising spins and S i denotes the i -th Ising spin. The correlation function q ( t ) seems to exhibit a nonexponential decay below the critical temperature of the nonrandom Ising model. An effective size ν of a cluster of ferromagnetically connected spins is defined as ν=(ln τ) 2 , where τ is the longest relaxation time in the cluster. It is found that the distribution function of ν behaves as P (ν)∝exp [-γν]. Although the asymptotic belaavior q ( t )∼exp [- C (ln t ) 2 ] is not reached in the time region studied by the Monte Carlo method, this distribution explains the long-time behavior of q ( t ).

Quantum corrections for inelastic atom-surface scattering.

We derive simple and exact expressions for the leading quantum corrections to the angular and energy distributions of an atom scattered inelastically from a solid crystalline surface. The quantum corrections are given in terms of classical trajectories and equilibrium correlation functions for the solid surface. Our results are useful when the incident atom has an energy of \ensuremath{\sim}1 eV and the surface has a temperature less than or comparable to the Debye temperature.

Generalized Ornstein-Zernike approach to many-particle equilibrium correlation functions

A new generalization of the Ornstein-Zernike equation for the functions describing equilibrium correlations between two groups of particles is proposed. The generalized direct correlation functions for two groups of particles are introduced, and the integral equations relating these functions to “full” correlations are derived. The derivation is based on diagrammatic analysis of the correlation functions. As an analogue of a cutting (nodal) vertex in the standard analysis leading to the Ornstein-Zernike equation we consider now a cutting set of vertices. Possible applications of the proposed formalism to the actual calculation of three- and four-particle equilibrium correlation functions are indicated.

Dissipation in nonlinear response

It is shown that the measurable nonlinear susceptibilities of third and higher orders probe only a fraction of equilibrium fluctuations, and in general can give only lower bounds on the equilibrium correlation functions. This contrasts with the linear and quadratic response, which completely determines the corresponding correlations by general fluctuation–dissipation theorems. The exactly solvable one-dimensional kinetic Ising model illustrates which fluctuations give rise to dissipation; only two-magnon processes of short wavelength and opposite momenta contribute to the third-order nonlinear susceptibilities, while two- and three-magnon processes contribute to the corresponding equilibrium correlation functions.

Linear kinetic theory of the square-well fluid

A recently proposed kinetic theory for a dense fluid of square-well particles is linearised and shown to satisfy time-reversal symmetry. The hydrodynamic modes and their extensions to short wavelengths are calculated with the aid of equilibrium correlation functions which are taken from molecular dynamics simulations and are presented as well. Under most conditions, potential energy fluctuations decay slowly in comparison with velocity fluctuations.

Transport Coefficients for Nonideal Hydrogen and Cesium Plasmas

AbstractThe transport coefficients of a nonideal plasma are expressed by equilibrium correlation functions within the ZUBAREV approach. Besides the treatment of the different interactions in a partially ionized plasma, this approach allows also for the currents of the heavy particles ions and atoms. Furthermore, the influence of an arbitrary mass ratio on the transport coefficients is investigated. Explicit results are shown for H (n= 1017… 1020cm−3,T= (1…3) · 104K) and Cs plasma (n= 1015…1021cm−3,T= (3…10) · 103K).

Three-particle equilibrium correlations in dense hard-sphere fluids.

We performed molecular-dynamics simulation experiments for a hard-sphere fluid at four high densities and determined the spatial Fourier transform of the three-particle equilibrium correlation function with two of the three particles at contact.

Linear kinetic theory of a suspension of interacting Brownian particles: I. Enskog renormalization

Time-dependent equilibrium correlation functions for interacting Brownian particles are analyzed. To this end the kinetic theory method recently proposed by Bławzdziewicz and Cichocki is used. The elementary processes (i.e. single particle diffusion and many-particle interactions) contributing to the evolution of the time-correlation functions are renormalized by the equilibrium correlations. The final result which corresponds to the Enskog renormalization in the kinetic theory of fluids is a convenient tool for an analysis of dynamical properties of dense suspensions.

Low-temperature and long-time anomalies of a damped quantum particle

The time evolution of a damped quantum particle is discussed. Dissipation is modeled by the bilinear coupling to a set of harmonic oscillators. Using a functional integral technique that accounts for initial correlations between the particle and the reservoir, one can express the dynamics of the damped particle entirely in terms of equilibrium correlation functions. The long-time behavior of these correlations is determined for memory damping arising from the coupling to a reservoir with spectral densityI(ω) ∞ωα at low frequencies, where α > 0. The time evolution of nonequilibrium initial states of the damped particle is discussed. At finite temperatures an initially localized state is found to spread subdiffusively or superdiffusively, depending on α. For α > 2 the damping becomes ineffective for long times, and the width of a state grows kinematically. At zero temperature and for α < 1, an initially localized state remains localized for all times. For α ≥ 1 the state spreads, but with a slower rate than at finite temperatures. Study of arbitrary initial states indicates that the process is ergodic at finite temperatures only for α ≤ 2 and at zero temperature for 1 ≤ α ≤ 2.

Thermodynamic limit in a charged plasma in equilibrium

We study the existence conditions for the thermodynamic limit in the chain of BBGKY equations for the equilibrium correlation functions of a charged plasma. It is shown that in order for the thermodynamic limit to exist the charge of the plasma cannot increase faster than the surface area of the plasma. When this condition is satisfied the equilibrium correlation functions of the charged plasma are asymptotically identical to the correlation functions of a neutral plasma in a self-consistent electrostatic field, which depends only on the one-particle correlation functions. For a plasma which is uniform everywhere except in a thin surface region, this field is found in explicit form. For a plasma occupying an infinite half-space, the problem is equivalent to a neutral plasma near a charged wall.

Thermodynamic basis for dielectric relaxation in complex materials.

Using the formalism of extended irreversible thermodynamics we study the time rate of change of the polarization vector in an arbitrary isotropic material. The corresponding time evolution equation is derived as well as its relationship with other relevant quantities in the system such as the heat flux. From these equations a polarization equilibrium correlation function is derived and the complex dielectric constant is also analyzed. We show how these quantities are related to those that have been obtained by several methods, some empirical and others based on microscopic models. Also the corresponding memory functions are obtained and shown to be exponentially decaying in time.

Microscopic theory of the transient grating experiment

We present the first fully microscopic theory of the generation of the transient grating signal from a crystal of interacting molecules. We derive a general expression for the signal, measured at a time immediately after a short probe pulse, in terms of the exciton Green function. This expression agrees with previous calculations of the grating signal, in which the experiment was treated as a diffraction process. It is also shown that the temporal profile of the signal pulse is, in general, different from that of the probe pulse. This result is inconsistent with the usual diffraction picture. Our treatment of this experiment as a transient four-wave mixing process rather than as a diffraction process allows us to derive an expression for the signal in terms of equilibrium correlation functions of the dipole operator. Using this formulation, we discuss the relationship of this experiment to other nonlinear spectroscopic techniques.

Equilibrium-fluctuation expression for the resistance of a Norton circuit.

In the familiar Green-Kubo expressions, transport coefficients are related to equilibrium fluctuations. The equilibrium ensemble which generates these fluctuations is obtained by setting the thermodynamic force to zero. For the special case of electrical conductivity, we show how the transport coefficient is related to fluctuations in the conjugate equilibrium ensemble where the thermodynamic flux, rather than the force, is set to zero. We show that the conjugate equilibrium correlation function so obtained is the memory function of the flux autocorrelation function.

Non-linear effects in rotational dynamics in the liquid state

Through general theoretical considerations, based only on the Markoffian assumption, it is shown that equilibrium non-gaussian properties can be used to predict how the decay of excited states deviates from a linear response theory behaviour. To check this prediction, a computer experiment on a ‘two-dimensional liquid’ of discs interacting via a Lennard-Jones plus electric dipole potential is carried out. The results obtained for both equilibrium and excited correlation functions confirm completely the intimate relation between equilibrium non-gaussian properties and excitation behaviour. Via approximate analytical expressions and exact solutions, it is also shown that a non-linear version of the popular itinerant oscillator provides the same equilibrium non-gaussian properties and, in consequence of that, the same kind of deviation from linear response as the two-dimensional system.

Linear response theory for thermoelectric transport coefficients of a partially ionized plasma

Abstract Electrical conductivity, thermopower and thermal conductivity for a partially ionized plasma are expressed within an extended Zubarev approach by equilibrium correlation functions. The Green function technique is used to evaluate the correlation functions in different approximations. Improvements of the Lenard-Balescu approximation are considered, which account for dynamical screening effects and higher Born approximations for the electron-electron, electron-ion and electron-atom interaction.

Asymptotic expansions for autocorrelation functions with Gaussian memory

Mori’s integro-differential equation for equilibrium correlation functions is investigated for a Gaussian memory. For this type of memory the first four moments of the autocorrelation function are reproduced exactly. So far, however, the underlying Mori equation has only been solved numerically. In this paper analytic short-time and long-time expansions are derived by the aid of which the behavior of the autocorrelation function can be described in the whole time domain. The analysis shows that there exists a critical parameter τc which separates the regime of oscillatory from that of nonoscillatory decay of the autocorrelation function. Finally, the Gaussian memory results are compared with those of an exponential memory.