Stochastic Dynamics in Irreversible Nonequilibrium Environments. 1. The Fluctuation−Dissipation Relation

Abstract. We investigate the fluctuation dynamics of a probe around a deterministic motion induced by interactions with driven particles. The latter constitute the nonequilibrium medium in which the probe is immersed and is modelled as overdamped Langevin particle dynamics driven by nonconservative forces. The expansion that yields the friction and noise expressions for the reduced probe dynamics is based on linear response around a time-dependent nonequilibrium condition of the medium. The result contains an extension of the second fluctuation-dissipation relation between friction and noise for probe motion in a nonequilibrium fluid. The problemWhether the environment of a system is in equilibrium or is driven into a nonequilibrium condition is important for characterizing internal motions and reactions. In other words the reduced system dynamics will reflect whether the environment is driven or not.For example, the relation between noise and friction on the system need no longer be described via the standard second fluctuation-dissipation (or Einstein) relation, we cannot unambiguously speak about entropy fluxes into the environment in terms of heat as there would be no Clausius relation, and system fluctuations or noise level in general are not simply quantified by a reservoir temperature. The present paper takes up the challenge of characterizing such an effective dynamics for a probe in contact with a nonequilibrium medium. One should have in mind that the medium consists of active elements or driven particles themselves in contact with a big thermal equilibrium reservoir (like surrounding air or water). Combined, the medium and the heat bath make up the nonequilibrium fluid. The system is called a probe here, but it can in general also refer to a collective or with j = 1, . . . , N labeling the particles and subject to independent standard white noises ξ j t . The β denotes the inverse temperature of the background equilibrium reservoir but the particles are effectively driven by the nonconservative force F . For simplicity we have not introduced explicitly friction and mass parameters, keeping only λ (coupling) and β as parameters. The particle interaction is given through the potential Φ while the potential U depends on the position X t of the probe, thus representing the back-reaction of the probe on each particle. Other and different types of interaction between probe and medium are possible, e.g. in terms of their center of mass coordinate with a mean field type potential U ( j x j t − X t ), here not discussed and inessential for the present level of discussion. Keeping to (1.1) the probe dynamics itself iswhere we indicate by K X t ,Ẋ t other aspects of the probe motion in the fluid. Its mass M is obviously also an important parameter for separation of time-scales between probe and fluid particle motion.The equations (1.1) and (1.2) starting at t = 0 are the basic evolution equations representing the coupled dynamics of probe and fluid particles. Nonequilibrium works directly on the medium which is ...

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“…The modification to the standard fluctuation-dissipation relation (cf. [10,2]) can be given in terms of…”

Section: Second Fluctuation-dissipation Relationmentioning

confidence: 99%

Friction and noise for a probe in a nonequilibrium fluid

Maes

Steffenoni

2015

Phys. Rev. E

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“…6 can be illustrated using the same model of the nonequilibirium change in the environment originally investigated in the context of the phenomenological iGLE. 16 To this end, numerical results are presented for the Hamiltonian system of a tagged free particle (V (q) = 0) bilinearly coupled to a bath of 200 harmonic modes whose smallest characteristic bath frequency is ω c = 1/(M τ ) where M = 4. Individual bath frequencies are taken at discrete values, ω i = (i − 1/2)ω c , and coefficients as per Eq.…”

Section: B the Free Particlementioning

confidence: 99%

“…All other parameters have identical values to those used in the numerical integration studies of the iGLE in Ref. 16 with the exception that 100,000 trajectories were used in this study, which is a 10-fold increase. In summary, all simulations share the following set of parameters: N = 100, 000, k B T = 2.0, γ 0 = 10.0, τ = 0.5, τ g = 0.2, ∆t = 1x10 −4 (t ≥ −8) and ∆t = 1x10 −3 (t < −8).…”

Section: B the Free Particlementioning

confidence: 99%

“…1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 In recent work, 16,17,18,19,20 it was suggested that in many instances, the environment is not stationary, and as such a nonstationary version of the GLE would be desirable. In principle, it is easy to write a nonstationary GLE aṡ…”

Section: Introductionmentioning

confidence: 99%

“…(1) and as such is an example of nonstationary stochastic dynamics. In recent work, 16,17,18,19,20 a generalization of this model was developed which includes arbitrary nonstationary changes in the strength of the friction, but like the xGLE model does not include changes in the response time. As such it is not quite the ultimately desired nonstationary GLE.…”

Section: Introductionmentioning

confidence: 99%

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An idealized model for nonequilibrium dynamics in molecular systems

Vogt

Hernandez

2005

The Journal of Chemical Physics

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The irreversible generalized Langevin equation (iGLE) contains a nonstationary friction kernel that in certain limits reduces to the GLE with space-dependent friction. For more general forms of the friction kernel, the iGLE was previously shown to be the projection of a mechanical system with a time-dependent Hamiltonian. [R. Hernandez, J. Chem. Phys. 110, 7701 (1999)] In the present work, the corresponding open Hamiltonian system is further explored. Numerical simulations of this mechanical system illustrate that the time dependence of the observed total energy and the correlations of the solvent force are in precise agreement with the projected iGLE.

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Hybrid QM/MM molecular dynamics simulations for an ionic S_N2 reaction in the supercritical water: OH⁻ + CH₃Cl → CH₃OH + Cl⁻

2002

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A hybrid real space quantum mechanical/molecular mechanical (RS-QM/MM) method has been applied to an ionic S(N)2 reaction (OH- + CH3Cl --> CH3OH + Cl-) in water solution to investigate dynamic solvation effects of the supercritical water (SCW) on the reaction. It has been demonstrated that the approaching process of OH- to methyl group is prevented by water molecules in the ambient water (AW), while the reaction takes place easily in the gas phase. Almost the same solvation effect on the dynamics of OH- is observed in the SCW, though the bulk density of water is substantially reduced compared with that of the AW. It has been shown that the solvation of the SCW around the OH anion is locally identical to that of the AW due to the strong ion-dipole interactions between OH- and water molecules. At the transition state, the QM/MM simulations have revealed that the excess electron is quite flexible, and the charge volume, as well as the fractional charges on atoms, vary seriously depending on the instantaneous solvent configurations. However, it has been found that the solvation energy in the SCW can be qualitatively related to the HOMO volume of the system by Born's equation.

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Stochastic Dynamics in Irreversible Nonequilibrium Environments. 1. The Fluctuation−Dissipation Relation

Cited by 51 publications

References 31 publications

Friction and noise for a probe in a nonequilibrium fluid

Friction and noise for a probe in a nonequilibrium fluid

An idealized model for nonequilibrium dynamics in molecular systems

Hybrid QM/MM molecular dynamics simulations for an ionic S_N2 reaction in the supercritical water: OH⁻ + CH₃Cl → CH₃OH + Cl⁻

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Stochastic Dynamics in Irreversible Nonequilibrium Environments. 1. The Fluctuation−Dissipation Relation

Cited by 51 publications

References 31 publications

Friction and noise for a probe in a nonequilibrium fluid

Friction and noise for a probe in a nonequilibrium fluid

An idealized model for nonequilibrium dynamics in molecular systems

Hybrid QM/MM molecular dynamics simulations for an ionic SN2 reaction in the supercritical water: OH− + CH3Cl → CH3OH + Cl−

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Product

Resources

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Hybrid QM/MM molecular dynamics simulations for an ionic S_N2 reaction in the supercritical water: OH⁻ + CH₃Cl → CH₃OH + Cl⁻