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Physical Review A

Stochastic variational approach to quantal-dissipative dynamical systems

Tetsuya Misawa

Published Sep 1, 1989• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.40.3387

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•Jan 1, 1994•1 citation
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.
Within the framework of stochastic calculus of variations for time-symmetric semimartingales X(t,ω), we consider two different stochastic versions of Maupertuis’ least action principle, in Lagrangian and Hamiltonian terms. The general results are applied to classical statistical mechanics, where they coincide with those of classical calculus of variations, and to Nelson’s stochastic mechanics, an approach to quantum mechanics where a time-symmetric semimartingale represents the position of a particle and the dynamics is expressed by a stochastic version of Hamilton’s principle of least action. Some historic examples of old quantum theory are discussed.
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 Lagrangian variational principle with the classical action leads, in stochastic mechanics, to Madelung’s fluid equations, if only irrotational velocity fields are allowed, while new dynamical equations arise if rotational velocity fields are also taken into account. The new equations are shown to be equivalent to the (gauge invariant) system of a Schrödinger equation involving a four-vector potential (A,Φ) and the coupled evolution equation (of magnetohydrodynamical type) for the vector field A. A general energy theorem can be proved and the stability properties of irrotational and rotational solutions investigated.
A canonical stochastic dynamical system for time-symmetric semimartingales is formulated by the stochastic least action principle in a new stochastic calculus of variations. A certain class of stochastic dynamical systems gives a Hamiltonian formalism of Nelson’s stochastic mechanics. In a manner analogous to classical mechanics, the notions of a stochastic Poisson bracket and canonical transformation are introduced to the stochastic dynamical systems. It is shown that the phase factor of the wave function plays the role of a generating function of the canonical transformation.
The stochastic quantization of dissipative systems is discussed. It is shown that in order to stochastically quantize a system with dissipation, one has to restrict the Fourier transform of the space-time variable to the positive half domain in the complex plane. This breaks the time-reversal invariance, which manifests in the formulation through the resulting noninvariant forms for the propagators. The relation of the stochastic approach with the Caldeira and Leggett path-integral method is also analyzed. † tzani@vana.physto.se; address after September 1994: Imperial College, Physics Dept., Prince Consort Road, London SW72BZThe quantization of a classical system may, in general, be hindered by several obstacles, such as nonrenormalizability, topological obstructions, anomalies etc. There exists, moreover, a whole class of systems for which there is no available method of quantization, namely non-hamiltonian systems. The quantization of systems with no lagrangian or hamiltonian formulation, apart from the wellknown examples of systems with dissipation, includes several interesting physical problems, such as the Bargmann-Wigner higher spin field equations [1] and field theories of fundamental anyons [2]. For the Bargmann-Wigner system of equations there is no action which directly reproduces this set of equations, although the quantization for higher spin systems has been done using more complicated and indirect ways [3]. The case of anyonic systems is known to be unsolved. The most common example of a non-hamiltonian system is, nevertheless, the case of a particle on the line with friction. Its equation of motion is m¨x + γ ˙x + dV dx = 0 (1) where overdot denotes time derivative and V (x) is the potential of the particle. If we assume an action of the form S = f(x, ˙x, t) dt (2) for the above equation, the variation of the action gives δS δx = dt ( ∂f ∂ ∂f ∂ ∂f