Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-orthogonal eigenvectors, fluctuation-dissipation relations and entropy production

Yan V. Fyodorov, Ewa Gudowska-Nowak, Maciej A. Nowak, Wojciech Tarnowski

Abstract

Celebrated fluctuation-dissipation theorem (FDT) linking the response function to time dependent correlations of observables measured in the reference unperturbed state is one of the central results in equilibrium statistical mechanics. In this Letter we discuss an extension of the standard FDT to the case when multidimensional matrix representing transition probabilities is strictly non-normal. This feature dramatically modifies the dynamics, by incorporating the effect of eigenvector nonorthogonality via the associated overlap matrix of Chalker-Mehlig type. In particular, the rate of entropy production per unit time is strongly enhanced by that matrix. We suggest, that this mechanism has an impact on the studies of collective phenomena in neural matrix models, leading, via transient behavior, to such phenomena as synchronization and emergence of the memory. We also expect, that the described mechanism generating the entropy production is generic for wide class of phenomena, where dynamics is driven by non-normal operators. For the case of driving by a large Ginibre matrix the entropy production rate is evaluated analytically, as well as for the Rajan-Abbott model for neural networks.

Non-orthogonal eigenvectors, fluctuation-dissipation relations and entropy production

Abstract

Celebrated fluctuation-dissipation theorem (FDT) linking the response function to time dependent correlations of observables measured in the reference unperturbed state is one of the central results in equilibrium statistical mechanics. In this Letter we discuss an extension of the standard FDT to the case when multidimensional matrix representing transition probabilities is strictly non-normal. This feature dramatically modifies the dynamics, by incorporating the effect of eigenvector nonorthogonality via the associated overlap matrix of Chalker-Mehlig type. In particular, the rate of entropy production per unit time is strongly enhanced by that matrix. We suggest, that this mechanism has an impact on the studies of collective phenomena in neural matrix models, leading, via transient behavior, to such phenomena as synchronization and emergence of the memory. We also expect, that the described mechanism generating the entropy production is generic for wide class of phenomena, where dynamics is driven by non-normal operators. For the case of driving by a large Ginibre matrix the entropy production rate is evaluated analytically, as well as for the Rajan-Abbott model for neural networks.
Paper Structure (6 sections, 51 equations, 3 figures)

This paper contains 6 sections, 51 equations, 3 figures.

Table of Contents

  1. Supplemental Material
  2. References

Figures (3)

  • Figure 1: Numerical results for $\tilde{\Delta}(\tau)$ for various values of non-normality parameter $\nu$. The transient behavior is dominant for intermediate time scales and is enhanced by larger non-normalities. The "anomalous" contribution to FDT decays at large times.
  • Figure 2: The activity as a function of time of each out of $N=40$ neurons in the linearized dynamics. In all three cases ($\nu=0, 20, 40$ ) eigenvalues are identical by construction. An onset of collective dynamics driven by the non-normality marker $\nu$ is clearly visible. All simulations started from exactly the same initial condition randomly chosen from the N-dimensional hypersphere. For simplicity, the ratios of excitatory to inhibitory neurons are identical.
  • Figure 3: Four largest eigenvalues of $C_0$ in Model 2 as a function of the non-normality parameter $\nu$. Only the largest one is affected, while the density of other eigenvalues remains almost unchanged.