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arXiv:hep-th/0309020 · Modern Physics Letters A

Classical brackets for dissipative systems

Giuseppe Bimonte, Giampiero V. M. Esposito, Giuseppe Marmo, C. Stornaiolo

Published Nov 20, 2003• 0 citations
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Abstract

We show how to write a set of brackets for the Langevin equation, describing the dissipative motion of a classical particle, subject to external random forces. The method does not rely on an action principle, and is based solely on the phenomenological description of the dissipative dynamics as given by the Langevin equation. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism.

DOI
10.1142/s0217732303012520

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We show how to write a set of brackets for the Langevin equation, describing the dissipative motion of a classical particle, subject to external random forces. The method does not rely on an action principle, and is based solely on the phenomenological description of the dissipative dynamics as given by the Langevin equation. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism.
The macroscopic description of a quantum particle with passive dissipation and moving in an arbitrary external potential is formulated in terms of the generalized Langevin equation. The coupling with the heat bath corresponds to two terms: a mean force characterized by a memory function \ensuremath{\mu}(t) and an operator-valued random force. Explicit expressions are given for the correlation and commutator of the random force. The random force is never Markovian. It is shown that \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(z), the Fourier transform of the memory function, must be a positive real function, analytic in the upper half-plane and with Re[\ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\omega}+i0+{0}^{+}0+)] a positive distribution on the real axis. This form is then derived for the independent-oscillator model of a heat bath. It is shown that the most general quantum Langevin equation can be realized by this simple model. A critical comparison is made with a number of other models that have appeared in the literature.
Phenomenological Equations of Motion for Dissipative Systems Lagrangian Hamiltonian and Hamilton-Jacobi Formulation of the Classical Dissipative Systems Noether's Theorem and Non-Noether Conservation Laws Dissipative Forces Derived from Many-Body Problems A Particle Coupled to a Field and the Damped Motion of a Central Particle Coupled to a Heat Bath Quantization of Dissipative Systems in General and of Explicitly Time-Dependent Hamiltonians in Particular Density Matrix and the Wigner Distribution Function for Damped Systems Path Integral Formulation of a Damped Harmonic Oscillator Quantization of the Motion of an Infinite Chain Heisenberg's Equations of Motion for a Particle Coupled to a Heat Bath Quantum Mechanical Models of Dissipative Systems and the Concept of Optical Potential.
The dissipative models in string theory can have more broad range of application: 1) Noncritical strings are dissipative systems in the "coupling constant" phase space. 2) Bosonic string in the affine-metric curved space is dissipative system. But the quantum descriptions of the dissipative systems have well known ambiguities. In order to solve the problems of the quantum description of dissipative systems we suggest to introduce an operator W in addition to usual (associative) operators. The suggested operator algebra does not violate Heisenberg algebra because we extend the canonical commutation relations by introducing an operator W of the nonholonomic quantities in addition to the usual (associative) operators of usual (holonomic) coordinate -momentum functions. To satisfy the generalized commutation relations the operator W must be nonassociative nonLieble (does not satisfied the Jacobi identity) operator. As the result of these properties the total time derivative of the multiplication and commutator of the operators does not satisfies the Leibnitz rule. This lead to compatibility of quantum equations of motion for dissipative systems and canonical commutation relations. The suggested generalization of the von Neumann equation is connected with classical Liouville equation for dissipative systems.
A quantum-classical bracket is proposed and is shown to satisfy the Jacobi identity, in contrast to previous definitions that obey this property only up to higher order terms in the Planck constant variant Planck's over 2pi. The Jacobi identity is required of a true Lie bracket and ensures that the Lie bracket of constants of motion is also a constant of motion. An explicit calculation of the Jacobi identity highlights the difference between the proposed and traditional definitions. A further example illustrates that the proposed bracket generates a more consistent quantum-classical dynamics than the traditional bracket. The traditional quantum-classical dynamics in the Henon-Heiles system diverges due to higher order variant Planck's over 2pi terms. The divergence is eliminated with the proposed bracket.
This chapter is devoted to the classical limit of the quantum dissipative action obtained in Chapter 3. We show how it yields Langevin, Fokker–Planck and optimal path descriptions of classical stochastic systems. These approaches are used to discuss activation escape, fluctuation relation, reaction models and other examples. Classical dissipative action In Section 3.2 we derived the Keldysh action for a quantum particle coupled to an Ohmic environment, Eq. (3.20). If only linear terms in the quantum coordinate X q (t) are kept in this action, it leads to a classical Newtonian equation with a viscous friction force, Eq. (3.21). Such an approximation completely disregards any fluctuations: both quantum and classical . Our goal now is to do better than that and to keep classical thermal fluctuations, while still neglecting quantum effects. To this end it is convenient to restore the Planck constant ħ in the action and then take the limit ħ → 0. For dimensional reasons, the factor ħ -1 should stay in front of the entire action. To keep the part of the action responsible for the classical equation of motion (3.21) free from the Planck constant it is convenient to rescale the quantum component as X q → ħ X q . Indeed, when this is done all terms linear in X q do not contain ħ. Finally, to have the temperature in energy units, one needs to substitute T with T /ħ.
The Brownian motion of a particle coupled to a heat bath can be derived from an exactly solvable many-body system. This motion is governed by the Langevin equation in Newtonian mechanics, and by an operator equation formally identical to the Langevin equation when the system composed of the particle plus the heat bath is quantized. To get this formal similarity between the classical and the quantal equations of motion of the particle, the classical dissipative system is formulated in terms of a modified Hamilton–Jacobi equation and is then quantized using the Schrödinger method. The result is identical with the Schrödinger–Langevin equation that has been obtained by quantizing the entire system and then isolating the motion of the particle. The non-linear wave equation describing the motion of a particle subject to conservative and time-dependent forces as well as frictional forces has been applied to the problems of motion of a wave-packet, and of the scattering and trapping of heavy-ions.