Information Dimension of Stochastic Processes on Networks: Relating Entropy Production to Spectral Properties

Mülken, Oliver; Sarah, Heinzelmann,; Dolgushev, Maxim

doi:10.1007/s10955-017-1785-z

“…This is readily understood since, as said above, the number of excited modes grows as a √ t. This fits with the general fact, that there exist a close relation between the growth of S and the spectrum of the master equation on networks (see e.g. [72] and the references therein) . If the spectral density of the operator shows scaling the entropy grows as ds 2 log t with time before reaching the equipartition value.…”

Section: Spectral Entropysupporting

confidence: 77%

“…As we have shown above, the energy transfer processes in the translation-invariant models occurs through a gradual, global, redistribution of the initial energy towards all the other modes. In the intermediate times, and assuming all the energy initially in one single mode E ν 0 (0) = δ ν,ν 0 for simplicity, one could thus perform a kind of "meanfield" approximation where [72]…”

Section: Spectral Entropymentioning

confidence: 99%

Thermalization of isolated harmonic networks under conservative noise

Lepri¹

2022

Preprint

0

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We study a scalar harmonic network with pair interactions and a binary collision rule, exchanging the momenta of a randomly-chosen couple of sites. We consider the case of the isolated network where the total energy is conserved. In the first part, we recast the dynamics as a stochastic map in normal modes (or action-angle) coordinates and provide a geometric interpretation of it. We formulate the problem for generic networks but, for completeness, also reconsider the translation-invariant lattices. In the second part, we examine the kinetic limit and its range of validity. A general form of the linear collision operator in terms of eigenstates of the network is given. This defines an action network, whose connectivity gives information on the out-of-equilibrium dynamics. We present a few examples (ordered and disordered chains and elastic networks) where the topology of connections in action spaces can be determined in a neat way. As an application, we consider the classic problem of relaxation to equipartition from the point of view of the dynamics of linear actions. We compare the results based on the spectrum of the collision operator with numerical simulation, performed with a novel scheme based on direct solution of the equation of motion in normal modes coordinates.

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“…The eigenvalue spectrum of the NT D trees is nonhomogenous. 30,32 It is dominated by the behavior of the linear chains characterized by the spectral dimension d s = 1. However, the smallest eigenvalues that describe large relaxation times (which correspond to the relaxation of the branches and subbranches as a whole [51][52][53] and related to large-scale characteristics such as gyration radius at the thermal equilibrium 45 ) march to a different tune.…”

Section: Nt D Treesmentioning

confidence: 99%

“…Under this approximation the integral can be readily computed, yielding a time dependence proportional to t α . This time regime holds for t τ 0 /λ we divide the integral of eqn (32) into three parts and use the small and long time behaviors of the Mittag-Leffler function discussed above,…”

Section: Analysis Of σ 2 (T)mentioning

confidence: 99%

“…This approach allows us to monitor explicitly the microscopic dynamics and to investigate its influence on the macroscopic behavior. In particular, we study treelike fractals (Vicsek fractals 25 ) and fractals with loops (dual Sierpiński gaskets 24 ), which both are rather homogeneous structures, the so-called NT D trees [30][31][32] that posses a heterogeneous density of states, and the small-world networks (SWNs) of ref. 33 whose density of states is characterized by the terminal value of the spectral dimension d s = 2.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Dynamics of networks in a viscoelastic and active environment

Grimm

¹

,

Dolgushev

²

2018

Self Cite

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We investigate the dynamics of fractals and other networks in a viscoelastic and active environment. The viscoelastic dynamics is modeled based on the generalized Langevin equation, where the activity is introduced to it by means of the exponentially correlated noise. The intramolecular interactions are taken into account by the bead-spring picture. The microscopic connectivity (studied in the form of Vicsek fractals, of dual Sierpiński gaskets, of NT trees, and of a family of deterministic small-world networks) reveals itself in the multiscale monomeric dynamics, which shows vastly different behaviors in the active and passive baths. In particular, the dynamics under active forces leads to a swelling that is characterized through power laws which are not present in the passive case. In all cases, the dynamics reflects the broad scaling behavior of the density of states and not necessarily the maximal relaxation time of the structures in a passive bath, as it is exemplified on the NT trees.

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Thermalization of Isolated Harmonic Networks Under Conservative Noise

Lepri

¹

2022

J Stat Phys

2

0

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We study a scalar harmonic network with pair interactions and a binary collision rule, exchanging the momenta of a randomly-chosen couple of sites. We consider the case of the isolated network where the total energy is conserved. In the first part, we recast the dynamics as a stochastic map in normal modes (or action-angle) coordinates and provide a geometric interpretation of it. We formulate the problem for generic networks but, for completeness, also reconsider the translation-invariant lattices. In the second part, we examine the kinetic limit and its range of validity. A general form of the linear collision operator in terms of eigenstates of the network is given. This defines an action network, whose connectivity gives information on the out-of-equilibrium dynamics. We present a few examples (ordered and disordered chains and elastic networks) where the topology of connections in action spaces can be determined in a neat way. As an application, we consider the classic problem of relaxation to equipartition from the point of view of the dynamics of linear actions. We compare the results based on the spectrum of the collision operator with numerical simulation, performed with a novel scheme based on direct solution of the equations of motion in normal modes coordinates.

show abstract

Information Dimension of Stochastic Processes on Networks: Relating Entropy Production to Spectral Properties

Cited by 6 publications

References 26 publications

Thermalization of isolated harmonic networks under conservative noise

Thermalization of isolated harmonic networks under conservative noise

Dynamics of networks in a viscoelastic and active environment

Thermalization of Isolated Harmonic Networks Under Conservative Noise

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