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

Variational principles for conservative and dissipative diffusions

R. Marra

Published Sep 15, 1987• 18 citations
Publisher

Abstract

A stochastic variational principle is formulated for the conservative diffusions of stochastic mechanics in terms of the classical action. A suitable class of variations is chosen taking into account the time-reversal invariance of the theory. The resulting equations of motion are the stochastic Euler-Lagrange equations. Moreover a derivation of the Navier-Stokes equation is presented.

DOI
10.1103/physrevd.36.1724

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A stochastic variational principle is formulated for the conservative diffusions of stochastic mechanics in terms of the classical action. A suitable class of variations is chosen taking into account the time-reversal invariance of the theory. The resulting equations of motion are the stochastic Euler-Lagrange equations. Moreover a derivation of the Navier-Stokes equation is presented.
We propose an extension to the relativistic case of the stochastic variational principles both of Lagrangian and Eulerian type. The action we use is the mean classical action evaluated on the paths of relativistic covariant diffusions. The resulting equations of motion are the relativistic stochastic Lagrange equations Nous proposons une extension au cas relativiste des principes variationnels de type lagrangien et eulerien. Nous utilisons comme action l'action moyenne classique evaluee sur les trajectoires des diffusions covariantes relativistes. Les equations du mouvement obtenues sont les equations de Lagrange stochastiques relativistes
In this paper we establish three variational principles that provide new foundations for Nelson’s stochastic mechanics in the case of nonrelativistic particles without spin. The resulting variational picture is much richer and of a different nature with respect to the one previously considered in the literature. We first develop two stochastic variational principles whose Hamilton–Jacobi-like equations are precisely the two coupled partial differential equations that are obtained from the Schrödinger equation (Madelung equations). The two problems are zero-sum, noncooperative, stochastic differential games that are familiar in the control theory literature. They are solved here by means of a new, absolutely elementary method based on Lagrange functionals. For both games the saddle-point equilibrium solution is given by the Nelson’s process and the optimal controls for the two competing players are precisely Nelson’s current velocity v and osmotic velocity u, respectively. The first variational principle includes as special cases both the Guerra–Morato variational principle [Phys. Rev. D 27, 1774 (1983)] and Schrödinger original variational derivation of the time-independent equation. It also reduces to the classical least action principle when the intensity of the underlying noise tends to zero. It appears as a saddle-point action principle. In the second variational principle the action is simply the difference between the initial and final configurational entropy. It is therefore a saddle-point entropy production principle. From the variational principles it follows, in particular, that both v(x,t) and u(x,t) are gradients of appropriate principal functions. In the variational principles, the role of the background noise has the intuitive meaning of attempting to contrast the more classical mechanical features of the system by trying to maximize the action in the first principle and by trying to increase the entropy in the second. Combining the two variational principles, we get the quantum Hamilton principle, i.e., a variational characterization of the logarithm of the wave function ψ. The Lagrangian is the Lagrangian of classical mechanics with the complex-valued velocity v−iu replacing the classical velocity. The dynamics is given by a stochastic differential equation for real-valued diffusions with complex-valued drift and driving noise processes. From the variational principle we derive a Newton-type law. We finally define the momentum process and show that its mean and variance coincide with those of the quantum momentum operator.
The stochastic variational principle which applies to diffusions and extremises the mean value of the classical action is generalised to include non-Markovian diffusions. A whole family of different stochastic actions, labelled by a time-dependent parameter, is obtained and all of them are shown to lead to the Shrodinger equation. As a result, the well known degeneracy of the class of stochastic processes which correspond to a single quantum state is recovered. It is also related to an arbitrariness in the choice of the noise which can occur in two places: in the definition of the noise, through the time-dependent parameter, and in the auto-correlations of the noise which, although constant during the variational method, remain undetermined. The whole scheme is given a covariant form.
A quantal-dissipative system described by the Schr\"odinger equation with an additional nonlinear term is formulated in the Lagrangian stochastic mechanics on the basis of the symmetric stochastic calculus of variations (SSCV). A certain class of stochastic Lagrangian dynamical systems defined through the least-action principle in SSCV is associated with solutions of the nonlinear Schr\"odinger equations, and thereby the quantal-dissipative system described by the equation is connected with a stochastic Lagrangian dynamical system. The long-time behavior of momentum, angular momentum, and energy, and the virial theorem for the stochastic Lagrangian system corresponding to the quantal-dissipative system, are investigated through the conserved quantities and the first variations of the functional induced by one-parameter transformations. Examples are given for a free particle and for the harmonic oscillator.
We consider stochastic variational principles for random processes taking values on a discrete configuration space. For a suitable time-reversal-invariant choice of the stochastic action, the resulting programming equations can be related to the Schr\"odinger equation for a discrete system. If the discrete system is considered as an approximation of a continuous system, then the limit reproduces Nelson's stochastic mechanics and allows one to derive the assumptions on the random noise acting on the system. In fact, the variational principle also gives information about the osmotic behavior of the process. Finally, we show that there are also critical processes whose behavior can be interpreted as a model for quantum measurement, because they relax in time to mixtures of processes. Therefore, stochastic variational principles can provide a very simple conceptual model simulating quantum behavior, both from the point of view of unperturbed time evolution and of the measurement phenomena leading to wave-function collapse.
The variational principle is one of important guiding principles in physics. Classical equations of motion of particle can be formulated so as to give the optimized path of an action. However, when there exist uncontrollable degrees of freedom such as noise, the optimized path is affected and the original classical equations of motion may not correspond to the optimized path. The stochastic variational method (SVM) is a framework to calculate the modified optimized path by the effect of noise. This method has been developed to show that the Schrödinger equation can be derived from the classical action which leads to Newton's equation of motion by taking into account the modification of the optimized path due to noise. In this work, we will extend this idea to the case of the continuum media and show that the Euler equation of the ideal fluid is converted to the Navier-Stokes equation or the Gross-Pitaevskii equation in SVM.