46 research outputs found
Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
Spectra of sparse non-Hermitian random matrices determine the dynamics of
complex processes on graphs. Eigenvalue outliers in the spectrum are of
particular interest, since they determine the stationary state and the
stability of dynamical processes. We present a general and exact theory for the
eigenvalue outliers of random matrices with a local tree structure. For
adjacency and Laplacian matrices of oriented random graphs, we derive
analytical expressions for the eigenvalue outliers, the first moments of the
distribution of eigenvector elements associated with an outlier, the support of
the spectral density, and the spectral gap. We show that these spectral
observables obey universal expressions, which hold for a broad class of
oriented random matrices.Comment: 25 pages, 4 figure
Universal transient behavior in large dynamical systems on networks
We analyze how the transient dynamics of large dynamical systems in the
vicinity of a stationary point, modeled by a set of randomly coupled linear
differential equations, depends on the network topology. We characterize the
transient response of a system through the evolution in time of the squared
norm of the state vector, which is averaged over different realizations of the
initial perturbation. We develop a mathematical formalism that computes this
quantity for graphs that are locally tree-like. We show that for unidirectional
networks the theory simplifies and general analytical results can be derived.
For example, we derive analytical expressions for the average squared norm for
random directed graphs with a prescribed degree distribution. These analytical
results reveal that unidirectional systems exhibit a high degree of
universality in the sense that the average squared norm only depends on a
single parameter encoding the average interaction strength between the
individual constituents. In addition, we derive analytical expressions for the
average squared norm for unidirectional systems with fixed diagonal disorder
and with bimodal diagonal disorder. We illustrate these results with numerical
experiments on large random graphs and on real-world networks.Comment: 19 pages, 7 figures. Substantially enlarged version. Submitted to
Physical Review Researc
Second Law of Thermodynamics at Stopping Times
Events in mesoscopic systems often take place at first-passage times, as is for instance the case for a colloidal particle that escapes a metastable state. An interesting question is how much work an external agent has done on a particle when it escapes a metastable state. We develop a thermodynamic theory for processes in mesoscopic systems that terminate at stopping times, which generalize first-passage times. This theory implies a thermodynamic bound, reminiscent of the second law of thermodynamics, for the work exerted by an external protocol on a mesoscopic system at a stopping time. As an illustration, we use this law to bound the work required to stretch a polymer to a certain length or to let a particle escape from a metastable state
Universal tradeoff relation between speed, uncertainty, and dissipation in nonequilibrium stationary states
We derive universal thermodynamic inequalities that bound from below the
moments of first-passage times of stochastic currents in nonequilibrium
stationary states and in the limit where the thresholds that define the
first-passage problem are large. These inequalities describe a tradeoff between
speed, uncertainty, and dissipation in nonequilibrium processes, which are
quantified, respectively, with the moments of the first-passage times of
stochastic currents, the splitting probability, and the mean entropy production
rate. Near equilibrium, the inequalities imply that mean-first passage times
are lower bounded by the Van't Hoff-Arrhenius law, whereas far from thermal
equilibrium the bounds describe a universal speed limit for rate processes.
When the current is the stochastic entropy production, then the bounds are
equalities, a remarkable property that follows from the fact that the
exponentiated negative entropy production is a martingale.Comment: 50 pages, 5 figure
Statistics of Infima and Stopping Times of Entropy Production and Applications to Active Molecular Processes
We study the statistics of infima, stopping times and passage probabilities
of entropy production in nonequilibrium steady states, and show that they are
universal. We consider two examples of stopping times: first-passage times of
entropy production and waiting times of stochastic processes, which are the
times when a system reaches for the first time a given state. Our main results
are: (i) the distribution of the global infimum of entropy production is
exponential with mean equal to minus Boltzmann's constant; (ii) we find the
exact expressions for the passage probabilities of entropy production to reach
a given value; (iii) we derive a fluctuation theorem for stopping-time
distributions of entropy production. These results have interesting
implications for stochastic processes that can be discussed in simple colloidal
systems and in active molecular processes. In particular, we show that the
timing and statistics of discrete chemical transitions of molecular processes,
such as, the steps of molecular motors, are governed by the statistics of
entropy production. We also show that the extreme-value statistics of active
molecular processes are governed by entropy production, for example, the
infimum of entropy production of a motor can be related to the maximal
excursion of a motor against the direction of an external force. Using this
relation, we make predictions for the distribution of the maximum backtrack
depth of RNA polymerases, which follows from our universal results for
entropy-production infima.Comment: 30 pages, 13 figure
Generic Properties of Stochastic Entropy Production
We derive an Ito stochastic differential equation for entropy production in
nonequilibrium Langevin processes. Introducing a random-time transformation,
entropy production obeys a one-dimensional drift-diffusion equation,
independent of the underlying physical model. This transformation allows us to
identify generic properties of entropy production. It also leads to an exact
uncertainty equality relating the Fano factor of entropy production and the
Fano factor of the random time, which we also generalize to non steady-state
conditions.Comment: 5 pages, 5 figures (contains Supplemental Material, 7 pages
Integral Fluctuation Relations for Entropy Production at Stopping Times
A stopping time is the first time when a trajectory of a stochastic
process satisfies a specific criterion. In this paper, we use martingale theory
to derive the integral fluctuation relation for the stochastic entropy production in a
stationary physical system at stochastic stopping times . This fluctuation
relation implies the law , which states
that it is not possible to reduce entropy on average, even by stopping a
stochastic process at a stopping time, and which we call the second law of
thermodynamics at stopping times. This law implies bounds on the average amount
of heat and work a system can extract from its environment when stopped at a
random time. Furthermore, the integral fluctuation relation implies that
certain fluctuations of entropy production are universal or are bounded by
universal functions. These universal properties descend from the integral
fluctuation relation by selecting appropriate stopping times: for example, when
is a first-passage time for entropy production, then we obtain a bound on
the statistics of negative records of entropy production. We illustrate these
results on simple models of nonequilibrium systems described by Langevin
equations and reveal two interesting phenomena. First, we demonstrate that
isothermal mesoscopic systems can extract on average heat from their
environment when stopped at a cleverly chosen moment and the second law at
stopping times provides a bound on the average extracted heat. Second, we
demonstrate that the average efficiency at stopping times of an autonomous
stochastic heat engines, such as Feymann's ratchet, can be larger than the
Carnot efficiency and the second law of thermodynamics at stopping times
provides a bound on the average efficiency at stopping times.Comment: 37 pages, 6 figure
Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems
We analyse the stability of large, linear dynamical systems of variables that
interact through a fully connected random matrix and have inhomogeneous growth
rates. We show that in the absence of correlations between the coupling
strengths, a system with interactions is always less stable than a system
without interactions. Contrarily to the uncorrelated case, interactions that
are antagonistic, i.e., characterised by negative correlations, can stabilise
linear dynamical systems. In particular, when the strength of the interactions
is not too strong, systems with antagonistic interactions are more stable than
systems without interactions. These results are obtained with an exact theory
for the spectral properties of fully connected random matrices with diagonal
disorder.Comment: 31 pages, 7 figure
Extreme value statistics of edge currents in Markov jump processes and their use for entropy production estimation
The infimum of an integrated current is its extreme value against the
direction of its average flow. Using martingale theory, we show that the infima
of integrated edge currents in time-homogeneous Markov jump processes are
geometrically distributed, with a mean value determined by the effective
affinity measured by a marginal observer that only sees the integrated edge
current. In addition, we show that a marginal observer can estimate a finite
fraction of the average entropy production rate in the underlying
nonequilibrium process from the extreme value statistics in the integrated edge
current. The estimated average rate of dissipation obtained in this way equals
the above mentioned effective affinity times the average edge current.
Moreover, we show that estimates of dissipation based on extreme value
statistics can be significantly more accurate than those based on thermodynamic
uncertainty ratios, as well as those based on a naive estimator obtained by
neglecting nonMarkovian correlations in the Kullback-Leibler divergence of the
trajectories of the integrated edge current.Comment: 38 pages, 6 figure
Localization and universality of eigenvectors in directed random graphs
Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the statistics of the right eigenvector components in directed random graphs with a prescribed degree distribution and with randomly weighted links. We obtain exact analytic expressions for the inverse participation ratio and show that right eigenvectors of directed random graphs with a small average degree are localized. Remarkably, if the fourth moment of the degree distribution is finite, then the critical mean degree of the localization transition is independent of the degree fluctuations, which is different from localization in undirected graphs that is governed by degree fluctuations. We also show that in the high connectivity limit the distribution of the right eigenvector components is solely determined by the degree distribution. For delocalized eigenvectors, we recover in this limit the universal results from standard random matrix theory that are independent of the degree distribution, while for localized eigenvectors the eigenvector distribution depends on the degree distribution