Equations of Motion in Nonequilibrium Statistical Mechanics

Robertson, Baldwin

doi:10.1103/physrev.144.151

Cited by 403 publications

(263 citation statements)

References 14 publications

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“…Although there exist other projection techniques as well, such as those in Refs. [48][49][50], the functional structure of the Zwanzig-Mori kinetic equations is independent of the considered complex system [51].…”

Section: Theoretical Description Of Memory Effects In Discrete Stochamentioning

confidence: 99%

Statistical quantifiers of memory for an analysis of human brain and neuro-system diseases

Demin

Yulmetyev

Panischev

et al. 2008

Physica A: Statistical Mechanics and its Applications

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On the basis of a memory function formalism for correlation functions of time series we investigate statistical memory effects by the use of appropriate spectral and relaxation parameters of measured stochastic data for neuro-system diseases. In particular, we study the dynamics of the walk of a patient who suffers from Parkinson's disease (PD), Huntington's disease (HD), amyotrophic lateral sclerosis (ALS), and compare against the data of healthy people (CO -control group). We employ an analytical method which is able to characterize the stochastic properties of stride-to-stride variations of gait cycle timing. Our results allow us to estimate quantitatively a few human locomotion function abnormalities occurring in the human brain and in the central nervous system (CNS). Particularly, the patient's gait dynamics are characterized by an increased memory behavior together with sizable fluctuations as compared with the locomotion dynamics of healthy patients. Moreover, we complement our findings with peculiar features as detected in phase-space portraits and spectral characteristics for the different data sets (PD, HD, ALS and healthy people). The evaluation of statistical quantifiers of the memory function is shown to provide a useful toolkit which can be put to work to identify various abnormalities of locomotion dynamics. Moreover, it allows one to diagnose qualitatively and quantitatively serious brain and central nervous system diseases.

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Section: Theoretical Description Of Memory Effects In Discrete Stochamentioning

confidence: 99%

Statistical quantifiers of memory for an analysis of human brain and neuro-system diseases

Demin

Yulmetyev

Panischev

et al. 2008

Physica A: Statistical Mechanics and its Applications

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“…It leads to the integro-differential equation with delay in time for the generalized canonical density. The quasiequilibrium projector (28) is more general than the projector obtained by Robertson [21] in the following sense: It is derived for any functional S with non-degenerate second differential D 2 Ψ S, for manifold of condition maxima of S and for any (nonlinear) differential equation. B. Robertson emphasized that this operator is non-Hermitian with respect to standard L 2 scalar product and in this sense is not a projector at all.…”

Section: Thermodynamic Projector Quasiequilibrium and Entropy Maximummentioning

confidence: 99%

“…In these cases the manifold of reduced description should be extended. We have a family of systems of moments M α = m α (Ψ), and a family of corresponding quasiequilibrium manifolds Ω α : The manifold Ω α consist of solutions of optimization problem S(Ψ) → max, m α (Ψ) = M for given α and all admissible values for M. To create a manifold of reduced description it is possible 3 In his dissertation [21] B. Robertson has studied "the equation of motion for the generalized canonical density operator". The generalized canonical density renders entropy a maximum for given statistical expectations of the thermodynamic coordinates.…”

Section: Thermodynamic Projector Quasiequilibrium and Entropy Maximummentioning

confidence: 99%

“…Of course, it can be applied to the systems considered in the memory function approach to project the initial equations onto the manifold of relevant distribution functions. Moreover, the particular case of the thermodynamic projector, namely the quasiequilibrium projector, was developed for this purpose by B. Robertson in the context of Nakajima -Zwanzig approach [21].…”

Section: Short Memory and Natural Projectormentioning

confidence: 99%

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Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

Gorban

Karlin

2004

Physica A: Statistical Mechanics and its Applications

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Three results are presented: First, we solve the problem of persistence of dissipation for reduction of kinetic models. Kinetic equations with thermodynamic Lyapunov functions are studied. Uniqueness of the thermodynamic projector is proven: There exists only one projector which transforms any vector field equipped with the given Lyapunov function into a vector field with the same Lyapunov function for a given ansatz manifold which is not tangent to the Lyapunov function levels.Second, we use the thermodynamic projector for developing the short memory approximation and coarse-graining for general nonlinear dynamic systems. We prove that in this approximation the entropy production increases. (The theorem about entropy overproduction.)In example, we apply the thermodynamic projector to derivation the equations of reduced kinetics for the Fokker-Planck equation. A new class of closures is developed, the kinetic multipeak polyhedra. Distributions of this type are expected in kinetic models with multidimensional instability as universally, as the Gaussian distribution appears for stable systems. The number of possible relatively stable states of a nonequilibrium system grows as 2 m , and the number of macroscopic parameters is in order mn, where n is the dimension of configuration space, and m is the number of independent unstable directions in this space. The elaborated class of closures and equations pretends to describe the effects of "molecular individualism". This is the third result.

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“…The present structure of the formalism consists in a vast extension and generalization of earlier pioneering approaches, among which we can pinpoint the works of Kirkwood [16], Green [17], MoriOppenheim-Ross [18], Mori [19], and Zwanzig [20]. NESEF has been approached from different points of view: some are based on heuristic arguments [18,[21][22][23][24], others on projection operator techniques [25][26][27] (the former following Kirkwood and Green and the latter following Zwanzig and Mori). The formalism has been particularly systematized and largely improved by the Russian School of statistical physics, which can be considered to have been initiated by the renowned Nicolai Nicolaievich Bogoliubov [28], and we may also name Nicolai Sergeivich Krylov [29], and more recently mainly through the relevant contributions by Dimitrii Zubarev [24,30], Sergei Peletminskii [22,23], and others.…”

Section: The Question Of Statistical Mechanics For Complex Structuredmentioning

confidence: 99%

Thermo-statistics of irreversible processes: a Boltzmann-Gibbs-style ensemble formalism

Luzzi¹,

Vasconcellos²,

Ramos³

2006

Braz. J. Phys.

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The area of Physics indicated in the title is nowadays of quite relevant interest, not only from the purely scientific point of view, but specially for its applied aspects associated to the present-time point-first-technologies. A particular research trend in the theory of irreversible processes, which are evolving in time in systems arbitrarily departed from equilibrium, is here briefly described. It consists in the construction of a Gibbs-style nonequilibrium ensemble formalism. The derivation of a nonequilibrium statistical operator is described (the variational approach of Predictive Statistical Mechanics is used). The main questions involved are presented and applications are briefly mentioned.

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Equations of Motion in Nonequilibrium Statistical Mechanics

Cited by 403 publications

References 14 publications

Statistical quantifiers of memory for an analysis of human brain and neuro-system diseases

Statistical quantifiers of memory for an analysis of human brain and neuro-system diseases

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

Thermo-statistics of irreversible processes: a Boltzmann-Gibbs-style ensemble formalism

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