Extended Irreversible Thermodynamics

Jou, David; Casas-Vázquez, José; Lebon, Georgy

doi:10.1007/978-3-642-97671-1

Cited by 551 publications

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“…-a, 05.60.Cd, 47.70. -n Linear flux-force dependences (such as Fick's law) are a special, limiting case of time-delayed equations [1], as first noticed by Maxwell [2]. Such equations arise, e.g., in time-delayed diffusion, which has been considered for many years [3][4][5] and applied to turbulent diffusion [3], optics in turbid media [6], x-ray bursters [7], etc.…”

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confidence: 99%

“…Parabolic reactiondiffusion equations have been applied to the spread of advantageous genes [8], dispersions of biological populations [9], epidemic models [10], etc. But, if information propagates at a finite speed, linear flux-force laws-and thus PRD equations-do not hold [1]. This unphysical feature can be avoided by making use of hyperbolic reaction-diffusion (HRD) equations [11,12].…”

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Time-Delayed Theory of the Neolithic Transition in Europe

1999

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The classical wave-of-advance model of the neolithic transition (i.e., the shift from hunter-gatherer to agricultural economies) is based on Fisher's reaction-diffusion equation. Here we present an extension of Einstein's approach to Fickian diffusion, incorporating reaction terms. On this basis we show that second-order terms in the reaction-diffusion equation, which have been neglected up to now, are not in fact negligible but can lead to important corrections. The resulting time-delayed model agrees quite well with observations. [ S0031-9007(98) [11,12]. Thus HRD equations, instead of the usual PRD equations, should be regarded as the first choice from a conceptual perspective. Hyperbolic reaction-diffusion equations have been very recently applied to the spread of epidemics [13], forest fire models [14], and chemical systems [15].An interesting application of PRD equations arose after archaeological data led to the conclusion that European farming originated in the Near East, from where it spread across Europe. The rate of this spread was measured [16], and a mathematical model was proposed according to which early farming expanded in the form of a PRD wave of advance [17]. Such a model provides a consistent explanation for the origin of Indo-European languages [18], and also finds remarkable support from the observed gene frequencies [19]. However, this PRD model predicts a velocity for the spread of agriculture that is higher than that inferred from archaeological evidence, provided that one accepts those values for the parameters in the model that have been measured in independent observations [17]. Here we will analyze this problem by means of a HRD model.Let p͑x, y, t͒ stand for the population density (measured in number of families per square kilometer), where x and y are Cartesian coordinates and t is the time. We assume that a well-defined time scale t between two successive migrations exists. We begin, as usual [20], noting that, between the values of time t and t 1 t, both migrations and population growth will cause a change in the number of families in an area differential ds dx dy, i.e., ͓p͑x, y, t 1 t͒ 2 p͑x, y, t͔͒ds ͓p͑x, y, t 1 t͒ 2 p͑x, y, t͔͒ m ds 1 ͓p͑x, y, t 1 t͒ 2 p͑x, y, t͔͒ g ds ,where the subindices m and g stand for migrations and population growth, respectively. We denote the coordinate variations of a given family during t by Dx and Dy. The effect of migrations on the evolution of p͑x, y, t͒ will be derived here by means of a simple extension of Einstein's model of Fickian diffusion [21]. The migration term in Eq. (1) can be written as ͓p͑x, y, t 1 t͒ 2 p͑x, y, t͔͒ m ds ds Z 12`Z 12`p ͑x 1 Dx, y 1 Dy, t͒f͑Dx, Dy͒dDx dDy 2 dsp͑x, y, t͒ ,where f͑Dx, Dy͒ is the fraction of those families lying at time t in an area ds, centered at ͑x 1 Dx, y 1 Dy͒, such that they are at time t 1 t in an area ds, centered at ͑x, y͒. Therefore, Z 12`Z 12`f ͑Dx, Dy͒dDx dDy 1 ,

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confidence: 99%

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confidence: 99%

Time-Delayed Theory of the Neolithic Transition in Europe

1999

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“…The average lifetime of laser modes is defined experimentally on spectra of absorption. The theory with lifetime generalizes the approaches by D.N.Zubarev [20,21], R.L.Stratonovich [52] and extended nonequilibrium thermodynamics [53]. The analogy to internal time [11] will be carried out.…”

Section: Discussionmentioning

confidence: 99%

“…all evolution of system is considered, and, contrary to the description of works [20,21], with the account of dissipative effects contained in k (1) . It is possible to compare expressions for <F m (t)> 1 , for example, for reverse temperature, to the similar expressions obtained in extended nonequilibrium thermodynamics [53]. Let's notice that it is possible to find analogues of the relations (and relations such as (7), (28) in a general view) in the common theory of statistical estimations and statistical decisions (for example, [54]).…”

Section: (Xt)=∇β(xt); X 1 (Xt)=-β(xt)∇v(xt); X I+1 (Xt)= -∇mentioning

confidence: 99%

Nonequilibrium Statistical Operator for Systems of Finite Size

Ryazanov¹

2012

IJTMP

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The lifetime of statistical system is introduced. It is supposed that the nonequilibrium statistical operator implicitly contains the lifetime. The operations of taking of invariant part, averaging on initial conditions used in works of D.N. Zubarev, temporary coarse-graining (Kirkwood), choose of the direction of time are replaced by averaging on lifetime distribution. The expression for lifetime change at transitions from quasi-equilibrium system to nonequilibrium one is derived. A sort of the nonequilibrium statistical operator for systems of the finite sizes is suggested.

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“…We have from (6)- (7) and (10) when At n=1 from correlations such as (11) it is possible to obtain the description of such nonequilibrium phenomena, as heat conductivity, mass transfer, the chemical reactions [4,5], close to the description by means of the Extended Irreversible Thermodynamics [17].…”

Section: Distribution For Nondisturbanced Lifetimementioning

confidence: 99%

Superstatistics and Lifetime

Ryazanov¹

2012

AJMS

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To describe the nonequilibrium states of a system we introduce a new thermodynamic parameter -the lifetime of a system. The statistical distributions which can be obtained out of the mesoscopic description characterizing the behaviour of a system by specifying the stochastic processes are written. Superstatistics, introduced as fluctuating quantities of intensive thermodynamical parameters, are obtained from statistical distribution with lifetime (random time to system degeneracy) as thermodynamical parameter (and also generalization of superstatistics). Necessary for this realization condition with expression for average lifetime of equilibrium statistical system obtained from stochastical storage model is consist. The obtained distribution passes in Gibbs or in superstatistics distribution depending on a measure of dissipativity in the system.

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Extended Irreversible Thermodynamics

Cited by 551 publications

References 0 publications

Time-Delayed Theory of the Neolithic Transition in Europe

Time-Delayed Theory of the Neolithic Transition in Europe

Nonequilibrium Statistical Operator for Systems of Finite Size

Superstatistics and Lifetime

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