Absolute continuity of projected SRB measures of coupled Arnold cat map lattices

In dissipative dynamical systems phase space volumes contract, on average. Therefore, the invariant measure on the attractor is singular with respect to the Lebesgue measure. As noted by Ruelle, a generic perturbation pushes the state out of the attractor, hence the statistical features of the perturbation and, in particular, of the relaxation, cannot be understood solely in terms of the unperturbed dynamics on the attractor. This remark seems to seriously limit the applicability of the standard fluctuation dissipation procedure in the statistical mechanics of nonequilibrium (dissipative) systems. In this paper we show that the singular character of the steady state does not constitute a serious limitation in the case of systems with many degrees of freedom. The reason is that one typically deals with projected dynamics, and these are associated with regular probability distributions in the corresponding lower dimensional spaces.

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“…Indeed, marginals of singular phase space measures, on spaces of sufficiently lower dimension than the phase space, are usually regular [11,12].…”

Section: Introductionmentioning

confidence: 99%

Fluctuation-dissipation relation for chaotic non-Hamiltonian systems

Colangeli¹,

Rondoni²,

Vulpiani³

2012

J. Stat. Mech.

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“…The uncoupled dynamics is simply defined to be (E(x), θ(x)) → (E(x), g(θ(x), E(x))) (1. 3) for each x ∈ Z d . The energies at each lattice site are thus conserved (as in the Hamiltonian case).…”

Section: Coupled Maps With a Conservation Lawmentioning

confidence: 99%

“…The θ dynamics is then local and the g dynamics has an invariant Sinai-Ruelle-Bowen measure ν 0 on N. In this case, it would be natural to prove that the E -dynamics is diffusive (in a sense to be specified below) a.s. in θ(0, ·) with respect to the measure ν = ν Z d 0 . For ψ = 0, if the θ-dependence of ψ is local and smooth and ψ small in a suitable sense [11,20,21,25,3,1,2,6,7] the θ dynamics still has an invariant SRB measure ν defined on the cylinder sets of N Z d . Then we want to prove diffusion a.s. with respect to ν.…”

Section: Coupled Maps With a Conservation Lawmentioning

confidence: 99%

Diffusion in Energy Conserving Coupled Maps

Bricmont

Kupiainen

2013

Commun. Math. Phys.

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Abstract. We consider a dynamical system consisting of subsystems indexed by a lattice. Each subsystem has one conserved degree of freedom ("energy") the rest being uniformly hyperbolic. The subsystems are weakly coupled together so that the sum of the subsystem energies remains conserved. We prove that the subsystem energies satisfy the diffusion equation in a suitable scaling limit.1. Coupled Maps with a Conservation Law 1.1. Diffusion in Hamiltonian Dynamics. One of the fundamental problems in deterministic dynamics is to understand the microscopic origin of diffusion. On a microscopic level, a physical system such as a fluid or a crystal can be modeled by Schrödinger or Hamiltonian dynamics, with a macroscopic number of degrees of freedom. Although the microscopic dynamics is not dissipative, dissipation should emerge in large spatial and temporal scales e.g. in the form of diffusion of heat or of concentration of particles.Dynamically, diffusion is related to the existence in the system of conserved quantities such as the energy which are extensive i.e. sums (or integrals) of local contributions that are 'almost conserved". Thus, if the system has a microscopic energy density E(t, x), x ∈ R d the total energy E tot = E(t, x)dx is a constant of motion but the energy density is, in general, not conserved since the dynamics redistributes it:The divergence acting on the energy current J guarantees conservation of the total energy. One would like to show that the conservative dynamics (1.1) turns, in a suitable scaling limit, to a diffusive one. Such a limit involves diffusive scaling of space and time, and taking typical initial conditions with respect to the Liouville measure with prescribed initial energy profile. The resulting macroscopic energy density should then satisfy a nonlinear diffusion equation of the typewhere κ(E) is the conductivity function.There has been a lot of numerical and theoretical work in recent years around these questions in the context of coupled dynamics. One considers a dynamical system consisting of a large number of elementary systems indexed by a subset V of a ddimensional lattice Z d . The total energy E of the system is a sum x∈V E(x) of energies E(x) which involve the dynamical variables of the system at lattice site x and

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“…There are several phenomena which are possible only when dynamics is higher dimensional. On the other hand, most of the existing studies in coupled map lattices are about coupled one-dimensional maps and a very few involving high dimensional maps, for example, coupled Chialvo's map, 12 coupled Henon map, 13 Arnold's cat map, 14 and modified Baker's map for Ising model. 15 Here we try to study coupled higher dimensional maps.…”

Section: Introductionmentioning

confidence: 99%

Dynamic phase transition from localized to spatiotemporal chaos in coupled circle map with feedback

Sonawane¹,

Gade²

2011

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We investigate coupled circle maps in the presence of feedback and explore various dynamical phases observed in this system of coupled high dimensional maps. We observe an interesting transition from localized chaos to spatiotemporal chaos. We study this transition as a dynamic phase transition. We observe that persistence acts as an excellent quantifier to describe this transition. Taking the location of the fixed point of circle map (which does not change with feedback) as a reference point, we compute a number of sites which have been greater than (less than) the fixed point until time t. Though local dynamics is high dimensional in this case, this definition of persistence which tracks a single variable is an excellent quantifier for this transition. In most cases, we also obtain a well defined persistence exponent at the critical point and observe conventional scaling as seen in second order phase transitions. This indicates that persistence could work as a good order parameter for transitions from fully or partially arrested phase. We also give an explanation of gaps in eigenvalue spectrum of the Jacobian of localized state.

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Absolute continuity of projected SRB measures of coupled Arnold cat map lattices

Cited by 15 publications

References 20 publications

Fluctuation-dissipation relation for chaotic non-Hamiltonian systems

Fluctuation-dissipation relation for chaotic non-Hamiltonian systems

Diffusion in Energy Conserving Coupled Maps

Dynamic phase transition from localized to spatiotemporal chaos in coupled circle map with feedback

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