Radu Balescu

On the covariant formulation of classical relativistic statistical mechanics

Abstract By developing Diracs ideas, it is shown that a clear, covariant formulation of relativistic statistical mechanics can be constructed explicity. The basis lies in the duality between the observers physical space-time which has the well-known Minkowski structure, and the dynamical systems phase space which has a completely different geometrical structure. The Lorentz transformations in physical space - expressed by the usual tensiorial formalism - are mapped in phase space by corresponding canonical transformations which form a representation of the Lorentz group. The generators of this representation are explicitly constructed in the case of a gas of charged particles interacting through the electromagnetic field. The latter is treated explicitly as a dynamical system. The equation for the Lorentz transformation of the distribution function in phase space is written explicitly; it is closely analogous to the Liouville equation. It can be solved exactly both for the free particles and for the free field, but not in the presence of interactions. In the latter case, it poses a problem which is quite analogous, and of the same degree of difficulty, as the solution of the ordinary Liouville equation. The whole theory has been presented in such a form as to exhibit the possibility of adaptation of the powerful modern many-body techniques to the problem of explicit Lorentz transformation of the dynamical quantities.

On the approach to non-equilibrium stationary states and the theory of transport coefficients

Abstract A general formula for the time dependent electric current arising from a constant electric field is derived along the line of ideas of Kubos theory. This formula connects the time dependence of the current to the singularities of the resolvant of Liouvilles operator of a classical system. It permits to make direct contact with the general theory of approach to equilibrium developed by Prigogine and his coworkers. It constitutes a framework for a diagram expansion of transport coefficients. A proof of the existence of a stationary state of its stability (to first order in the field) are given. It is rigorously shown that, whereas the approach to the stationary state is in general governed by complicated non-markoffian equations, the stationary state itself (and thus the calculation of transport coefficients) is always determined by an asymptotic cross section. This implies that transport coefficients can always be calculated from a markoffian Boltzmann-like equation even in situations in which that equation does not describe properly the approach to the stationary state.

On the structure of the time-evolution process in many-body systems

Abstract The results of the Prigogine-George-Henin theory of “subdynamics” are extended to cover more general systems, such as spatially inhomogeneous systems or relativistic systems. The theory is presented in an abstract form, from which any particular case can be obtained by using an appropriate realization of the mathematical symbols. A number of new results are obtained in this way. The internal symmetry of the theory is clearly emphasized in the present formalism.

A unified formulation of the kinetic equations

Abstract It is shown that a formulation of non-equilibrium statistical mechanics is possible in such a way as to treat symmetrically homogeneous and inhomogeneous systems, vacuum and correlations. These special cases needed distinct treatment in hitherto developed formalisms. The new formulation makes extensive use of so-called “projection operators”. These are defined abstractly by using a minimum of assumptions. Very compact equations are obtained by using this formalism; these equations contain all previously known results as special cases. However, it is shown that the abstract “projection operators” also allow for realizations more general than the ones used hitherto. These realizations are important in the study of inhomogeneous systems of coupled oscillators (or fields). It is also shown that the conventional name “projection operators” is a quite improper one. The symmetry of our formulation is essential for the study of certain problems in which the usual decompositions have no intrinsic meaning (such as the study of relativistic invariance).

Statistical mechanics of a spin-polarized plasma

The statistical mechanics of a spin-polarized plasma is investigated in detail. A rigorous quantum-mechanical description is constructed in terms of a generalized matrix Wigner function. In order to ensure the manifest gauge invariance of the theory, the non-canonical variables q (position) and π (mechanical momentum) are used for the particles. The evolution equation for the phase-space Wigner function, as well as the BBGKY hierarchy for the reduced distribution functions, are derived. A general expression is found for the quantum-mechanical realization of the Lie bracket of any pair of dynamical functions. In the quasi-classical limit, the equations of evolution and the Lie bracket reduce to a simple form. Our approach is compared with the previous semi-phenomenological theory of Cowley, Kulsrud and Valeo.

Relativistic statistical thermodynamics

Abstract The formalism of relativistic statistical mechanics, developed in previous papers, provides a very straightforward proof of the Lorentz invariance of the canonical equilibrium distribution function. This theorem automatically determines the Lorentz transformation law of the temperature and of the free energy, and hence of all the thermodynamic functions. The transformation rule for the internal energy is discussed in great detail. It is shown that for a system of finite size the energy does not transform as the fourth component of a 4-vector, as is assumed by some authors. The boundary effect is responsible for this non-vectorial character. On the contrary, local quantities such as the energy density, for which the boundaries play no role, have a tensorial character. The controversy which arose in the recent literature about these problems is discussed in detail, with an emphasis on the sources of ambiguity. It is shown that, besides the canonical distribution, there exists an infinite number of Lorentz-invariant equilibrium distributions for each value of the velocity; all these functions reduce to the usual canonical distribution in the rest frame. Each one leads to a different thermodynamic formalism, special cases of which are the formalisms suggested by Ott and by Landsberg. All these formalisms can be reduced to the same form-invariant of Planck by means of a “gauge transformation” of the temperature and of the free energy. Strict equilibrium statistical mechanics cannot determine a unique choice of the absolute gauge. If such a determination has any meaning at all, it can only come from an extension of this investigation to cover non-equilibrium processes.

Lorentz transformations in phase space and in physical space

Abstract The relation between “tensorial” Lorentz transformations in physical space and “canonical” Lorentz transformations in phase space is examined in detail. It is shown in particular that the Lorentz-Liouville equation transforms the phase-space distribution function in such a way that the relevant average values computed with it, transform precisely as vectors or tensors. This has been proved explicitly in the case of the current-density vector and the energy-momentum tensor. An important fact which comes out of the theory is that the tensorial variance is not attached intrinsically to a given function but is determined by the dynamical nature of the system For instance, the average identified with the energy-momentum tensor for free particles, transforms as a tensor in a free particle system, but not in a system of interacting particles.

V-Langevin equations, continuous time random walks and fractional diffusion

The following question is addressed: under what conditions can a strange diffusive process, defined by a semi-dynamical V-Langevin equation or its associated hybrid kinetic equation (HKE), be described by an equivalent purely stochastic process, defined by a continuous time random walk (CTRW) or by a fractional differential equation (FDE)? More specifically, does there exist a class of V-Langevin equations with long-range (algebraic) velocity temporal correlation, that leads to a time-fractional superdiffusive process? The answer is always affirmative in one dimension. It is always negative in two dimensions: any algebraically decaying temporal velocity correlation (with a Gaussian spatial correlation) produces a normal diffusive process. General conditions relating the diffusive nature of the process to the temporal exponent of the Lagrangian velocity correlation (in Corrsin approximation) are derived. It is shown that a bifurcation occurs as the latter parameter is varied. Above that bifurcation value the process is always diffusive.

Langevin equation versus kinetic equation: Subdiffusive behavior of charged particles in a stochastic magnetic field

The running diffusion coefficient D(t) is evaluated for a system of charged particles undergoing the effect of a fluctuating magnetic field and of their mutual collisions. The latter coefficient can be expressed either in terms of the mean square displacement (MSD) of a test particle, or in terms of a correlation between a fluctuating distribution function and the magnetic field fluctuation. In the first case a stochastic differential equation of Langevin type for the position of a test particle must be solved; the second problem requires the determination of the distribution function from a kinetic equation. Using suitable simplifications, both problems are amenable to exact analytic solution. The conclusion is that the equivalence of the two approaches is by no means automatically guaranteed. A new type of object, the ‘‘hybrid kinetic equation’’ is constructed: it automatically ensures the equivalence with the Langevin results. The same conclusion holds for the generalized Fokker–Planck equation. The (B...

Dynamical correlation patterns: A new representation of the Liouville equation

Abstract The reduced distribution functions characterizing a many-particle system are grouped together into a set called “distribution vector”. The BBGKY hierarchy then appears as a single matrix equation, algebraically equivalent to the Liouville equation. Each reduced distribution function is further decomposed into a “dynamical” cluster representation. Contrary to the usual representation, based on the functional form of the various terms (factorization), the new “correlation patterns” are defined by using a purely dynamical criterion: separate equations of evolution are written for the correlation patterns. These equations can again be viewed as components of the single (matrix) Liouville equation, in the correlation-patterns representation. This representation has been devised in order to be combined with the modern, very powerful methods of non-equilibrium statistical mechanics. It provides a concrete realization of the formerly developed abstract theory. There is no difficulty with the thermodynamic limit, which can be taken from the very beginning. The method is particularly well adapted to the study of inhomogeneous systems, in which case it provides a simple and clear-cut separation of inhomogeneity and correlation effects. The relation between usual (“static”) and dynamic correlation patterns is studied. Finally, a diagram technique, suitable for practical calculations, is constructed.