Time: Please see below and follow announcements.
Location: tba
General Idea: The aim of these meetings is to provide a forum to discuss both our own work and other recent (or not-so-recent) developments in statistical mechanics. This is intended to be a "study group" not a "seminar"; speakers are asked to proceed slowly and audience participation is actively encouraged. After each talk there is also the opportunity for further informal discussions in more relaxed surroundings(link is external).
Schedule: The list below gives a provisional outline of talks planned for this semester. Please note, however, that our schedule is fairly flexible to allow for the evolving interests of the group.
Organizers: Vicenzo Nicosia; if you want to be on the group's mailing list, or perhaps want to give a talk yourself, please email us (v.nicosia@qmul.ac.uk).
It is a rather common belief that the only probability distribution occurring in the statistical physics of many-particle systems is that of Boltzmann and Gibbs (BG). This point of view is too limited. The BG-distribution, when seen as a function of parameters such as the inverse temperature and the chemical potential, is a member of the exponential family. This observation is important to understand the structure of statistical mechanics and its connection with thermodynamics. It also is the starting point of the generalizations discussed below. Recently, the notion of a generalized exponential family has been introduced, both in the mathematics and in the physics literature. A sub-class of this generalized family is the q-exponential family, where q is a real parameter describing the deformation of the exponential function. It is the intention of this talk to show the relevance for statistical physics of these generalizations of the BG- distribution. Particular attention will go to the configurational density of classical mono-atomic gases in the micro- canonical ensemble. These belong to the q-exponential family, where q tends to 1 as the number of particles tends to infinity. Hence, in this limit the density converges to the BG-distribution.
We consider some peculiarities of subdiffusive transport within the continuous time random walk (CTRW) model as appearing in the mean-field description of particles' motion in random potentials (energetic disorder). The anomalous diffusion under CTRW is a process with non-stationary increments. This non-stationarity introduces explicit dependence of observables on the time elapsed from preparing the system in its present state, and corresponds to aging of the process. Aging leads to such unusual properties of the system's time evolution as death of linear response to an external stimulus or as intrinsic ergodicity breaking. The last can have different manifestations, like the explicit dependence of the moving time averages on the interval of averaging or like universal fluctuations in time-averaged kinetic coefficients whose ensemble averages are sharp. These properties lead to several interesting effects, which are specific for energetic disorder and which can be used for distinguishing this mechanism of anomalous subdiffusion from other possible mechanisms (like the existence of slow modes or diffusion in geometrically disordered systems).
The unification of relativity and thermodynamics has been a subject of considerable debate over the last 100 years. The reasons for this are twofold: (i) Thermodynamic variables are nonlocal quantities and, thus, single out a preferred class of hyperplanes in spacetime. (ii) There exist different, seemingly equally plausible ways of defining heat and work in relativistic systems. These ambiguities led, for example, to various proposals for the Lorentz transformation law of temperature. Traditional "isochronous" formulations of relativistic thermodynamics are neither theoretically satisfactory nor experimentally feasible. I will discuss how these deficiencies can be resolved by defining thermodynamic quantities with respect to the backward-lightcone of an observation event. This approach also allows for a straightforward extension of thermodynamics to general relativity. Theoretical considerations are illustrated through simple relativistic many-body simulations.
Non-equilibrium systems have long-ranged spatial correlations even far away from critical points. These correlations have been observed experimentally, and recently it has been shown that they lead to nonlocal large deviation functionals in some models of heat and mass transport.
In this talk, we take a new point of view to non-equilibrium correlations. We discuss a functional level inverse problem, in which the state of a fluctuation field is estimated from a small amount of spatial information. In practice, this is accomplished by observing a dilute marker in a stationary flow. The particular problem we discuss is the estimation of the structure of an underlying medium, which determines the rate of transport. This system has the same kind of correlation structure as some driven diffusive systems, and which is observed in a Rayleigh-Bénard system. Thus the methods applied in media estimation could be useful in state estimation of time-dependent fluctuation fields.
We show how a synthetic gene network can function, in an optimal window of noise, as a robust logic gate. Interestingly, noise enhances the reliability of the logic operation. Further, we consider a two-dimensional model of a gene network, where we show how two complementary gate operations can be achieved simultaneously. We generalize this idea in two dimensional dynamical systems to achieve any two combinations of AND,OR, and XOR gates in parallel.
We consider diffusion-like equations with time and space fractional derivatives of distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose mean squared displacement does not change as a power law in time. Correspondingly, the underlying processes cannot be viewed as self-affine random processes possessing a unique Hurst exponent. We show that different forms of distributed-order equations, which we call 'natural' and 'modified' ones, serve as a useful tool to describe the processes which become more anomalous with time (retarding subdiffusion and accelerated superdiffusion) or less anomalous demonstrating the transition from anomalous to normal diffusion (accelerated subdiffusion and truncated Lévy flights). Fractional diffusion equation with the distributed-order time derivative also accounts for the logarithmic diffusion (strong anomaly).
The large-deviation properties of different types of random graphs, are studied using numerical simulations. Also an application to the sequence-alignment problem and to the ground state calculation of spin glasses is given.
First, distributions of the size of the largest component, in particular the large-deviation tail, are studied numerically for two graph ensembles, for Erdoes-Renyi random graphs with finite connectivity and for two-dimensional bond percolation. Probabilities as small as 10^-180 are accessed using an artificial finite-temperature (Boltzmann) ensemble and applications of the Wang-Landau algorithm. The distributions for the Erdoes-Renyi ensemble agree well with previously obtained analytical results. The results for the percolation problem, where no analytical results are available, are qualitatively similar, but the shapes of the distributions are somehow different and the finite-size corrections are sometimes much larger. Furthermore, for both problems, a first-order phase transition at low temperatures T within the artificial ensemble is found in the percolating regime, respectively.
Second, the some recent results for distributions of the diameter are presented and compared to partial analytic results which are available from previous studies for Erdoes-Renyi random graphs in the small connectivity region.
Finally, large-deviation properties of the distribution of sequence alignment scores of proteins (using the standard database parameters with (12,1) affine gap costs and BLOSUM score matrix) and the distribution of ground-state energies for the mean-field (Sherrington-Kirkpatrick) spin glass are presented.
I provide a macrostatistical treatment of hydrodynamical fluctuations about nonequilibrium steady states of reservoir driven many-particle systems, which may be classical or quantal. The treatment is centred on the dynamics of locally conserved macroscopic observables, subject to very general assumptions of local equilibrium, chaoticity and an extension of Onsager's regression hypothesis to situations that may be far from global equilibrium. On this basis, I establish that the hydrodynamical fluctuations execute a classical Markov process, that is completely determined by the macroscopic properties of the system. Furthermore, the structure of this process carries generalisations of both Onsager's irreversible thermodynamics and Landau's fluctuating hydrodynamics to systems that may be far from thermal equilibrium.
We will present the recently developed theory of hypocoercivity, which helps to prove exponential convergence to equilibrium in many cases of interest. Also, we will show how to find the exact rate of exponential convergence to equilibrium for (quadratic) hypoelliptic operators. As an example, we will apply these methods to one of the possible Markovian approximations of the non-Markovian Langevin equation.
When the weather is fine, then you know it's the time
For messing about on the river
(Tony Hatch)
So goes the song, but how do we know? Catchment hydrology is the application of basic physical laws (such as mass conservation) and hydrodynamic theory, at the scale of river basins. It is of vital importance in water resources management and the forecasting and mitigation of floods, a natural hazard which costs the UK £2.5 billion every year: a figure which is likely to increase in the future due to greater urbanisation and climate change. It is inherently a multi-disciplinary and applied subject, with active research ranging from the underlying mathematics to shaping government policy. I will present an overview of the work we do at CEH in this area, with emphasis on the various mathematical models employed and the extensive data resources we curate, many of which are publicly available through our online information gateway.
Microtubules are highly dynamic biopolymer filaments involved in a wide variety of biological processes like cell division and intracellular transport. These filaments are semi-flexible polymers, i.e. their bending energy is comparable to the thermal energy. Even though they form a rather stiff and highly cross-linked structural network, it has been shown that they typically exhibit significant bends on all length scales in the living cell.
Measurements of thermally driven microtubule fluctuations under laboratory conditions reveal a persistence length, which is several orders of magnitude larger than observed in living cells. Several studies investigated the interactions of motor proteins and mircotubules. Experiments and in vivo observations have shown microtubules to exhibit buckling instabilities induced by molecular motors, which compress the filament longitudinally. However, direct transversal motor activity on microtubules cannot be ruled out.
I will present a toy model for transversal deformations of a microtubule due to active processes. Simulations of motor proteins deforming a microtubule against a background network mimic the microtubule's behaviour under rapid step-like force fluctuations. The analysis of these fluctuations reveal interesting aspects of the apparent persistence length, motor cooperation as well as global filament displacement, which can be interpreted as a fractional random walk.
We construct a one-parameter family of real analytic uniformly expanding circle maps $f_{\lambda}, \lambda \in (0,1)$ for which the eigenvalues of the corresponding Perron-Frobenius operator acting upon analytic functions are given by $\{\lambda^k\}_{k \in \mathbb{Z_+}}$
Introduction:
The Tangled Nature Model of evolution is an individual based, stochastic model, which describes, with good agreement with actual observations, the evolution of a simple ecology. Its most remarkable feature is that its dynamics alternates between periods of meta-stable configurations and periods of hectic transitions, where the model does not show clear occupancy patterns and the population is spread randomly across the type space.
Hypothesis and methods:
The aim of this project is to analyze the stability of the stable configurations (qESS states) shown in the model by using a dynamical system approach. Indeed, we can derive a deterministic system of equations which approximates the dynamics of the model. Clearly, we can analyze the local stability of its fixed points by linearizing the equations about the equilibrium configurations. The idea in this work is to run simulations of the stochastic model to obtain specific configurations of the qESS states and use their averaged occupancy of these configurations to calculate the linearized dynamical matrix. The eigenvalues of this matrix are expected to be able to give useful information about the stability of the meta-stable states. In this presentation we will describe in details the ideas introduced and describe the results obtained.
In this talk, I will present three different pieces of work, unified by a common theme, that of memory.
The first involves the modelling of eyetracking data – specifically, we have analysed the visual movements of sample populations subjected to simultaneous visual and aural inputs. We looked for correlations between these two forms of sensory stimuli via the analysis of the probability distributions of saccades and fixations. As our sample populations involved literate as well as illiterate people, we were able to investigate the effect of literacy on cognitive processing. This was particularly manifest in the case of fixations, where it appears that literacy leads to the presence of a characteristic (attentional) time scale in the appropriate probability distribution. On the other hand, scale-invariance is observed in the saccadic distributions, independent of the literacy level of the subjects. We suggest that these are characterised by Levy-like dynamics.
Another piece of work involves the role of synaptic metaplasticity to model the separate storage of long- and short-term memories in the human brain. We have presented and analysed two models of metaplastic synapses, whose main difference lies in the effect of a contrarian event on long-term memories. In one model, the effect is to build up an opposite memory of similar depth, while in the other, the effect is more short-term. Although the transient properties of the models reflect this difference, their asymptotic behaviour is robustly the same – power-law forgetting with the same universal exponent, is manifested.
A third research area involves that of game-theoretic formulations of synaptic plasticity. The main motivation for this work is that competitive dynamics are thought to occur in many processes of learning involving synapses. We have shown that the competition between synapses in their weak and strong states gives rise to a natural framework of learning, with the prediction of memory inherent in a timescale for forgetting a learned signal. Among our main results is the prediction that memory is optimized if the weak synapses are really weak, and the strong synapses are really strong. We have also studied the dynamic responses of the effective system to various signal types, particularly with reference to an existing empirical motor adaptation model. The dependence of the system-level behaviour on the synaptic parameters and the signal strength has been analysed with a view to optimal performance, and illustrates the functional role of multiple timescales.
Evolutionary dynamics have been traditionally studied in infinitely large homogeneous populations where each individual is equally likely to interact with every other individual. However, real populations are finite and characterised by complex interactions among individuals. Over the last few years there has been a growing interest in studying evolutionary dynamics in finite structured populations represented by graphs. An analytic approach of the evolutionary process is possible when the contact structure of the population can be represented by simple graphs with a lot of symmetry and lack of complexity. Such graphs are the complete graph, the circle and the star graph. Moreover, this is usually infeasible on complex graphs and the use of various assumptions and approximations is necessary for the exploration of the process. We propose a powerful method for the approximation of the evolutionary process in populations with a complex structure. Comparisons of the predictions of the model constructed with the results of computer simulations reveal the effectiveness of the process and the improved accuracy that it provides when compared to well-known pair approximation methods.
The asymmetric simple exclusion process (ASEP) is one of the simplest, and yet one of the most studied models in non-equilibrium statistical physics. It is also related, more or less closely, to problems in biophysics (such as ribosomes moving on a m-RNA, which is what it was originally meant to describe), growing interfaces, pedestrian and car traffic, quantum spin chains, and many more.
After a brief reminder on the theory of large deviations, I will show how, by using a method based on Derrida, Evans, Hakim and Pasquier's matrix Ansatz, one can obtain the exact fluctuation statistics of the current of particles that characterises the steady state of the ASEP, for any finite size and any values of the parameters. I will then analyze the behaviour of these fluctuations in the large size limit. Finally, if time allows, I will discuss what other quantities we might be able to access using this method.
Refs : J. Phys. A: Math. Theor. 44 (2011) 315001 , Phys. Rev. Lett. 109, 170601 (2012)
The model proposed by Y. Kuramoto in 1975 has been widely used since then to study synchronous patterns in globally-connected biological, technological and social systems, and has been recently extended to systems of oscillators coupled through heterogeneous topologies. It has been found that the emergence of synchronised states in networks of coupled oscillators crucially depends on the topological structure of the underlying coupling graph. We consider a frustrated Kuramoto model in which the oscillators are coupled through a complex networks and have identical natural frequencies, but only phase-lock synchronisation is attainable. We show that the presence of symmetries in the coupling network has a central role on the synchronisation of the system. In particular we found that, at the stationary state, any two symmetric nodes of the graph end up having identical phases, i.e. they are perfectly synchronised, and this happens despite the distance of the two nodes in the graph. We prove that this remote synchronisation is induced solely by network symmetries, and we discuss an application to the human brain.
Determining the topology of a network is relevant for assessing network stability, dynamics and function. However, many surveyed networks that are reported in public data repositories, as protein interaction networks, are imperfect and often biased samples of the true underlying networks. This observation poses the interesting question of how representative a random subnet is for the global network and whether extrapolation of network topologies from partial network data to the whole network can be done. We use random graph ensembles tailored to real networks to compare network topologies macroscopically and quantify in a precise and practical way the effects of sampling on networks. We perform a systematic study of the effects of sampling on topological features of large protein interaction networks for a broad family of sampling protocols that include random and connectivity dependent node and/or link undersampling and oversampling, and derive exact formulae for degree distributions and degree correlation kernels of sampled networks, in terms of those of the underlying true network. Our formula suggest that inference on the topology of the true underlying network can be done accurately.
We consider the formation of large scale structures (zonal jets and vortices), in planetary atmospheres. We will prove that modern statistical mechanics approaches predict the outcome of the very complex dynamics of geostrophic turbulence and predict jet and vortex structures and shapes.
Based on the equilibrium statistical mechanics of the quasi-geostrophic dynamics, we will discuss a model of the Great Red Spot of Jupiter, and of other Jovian vortices. We will discuss the non-equilibrium statistical mechanics of Jupiter jets.
Large deviation theory is the basic mathematical tool on which those results are built. We will discuss the relations between large deviation and fluid mechanics at a basic level.
Many growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties. Such structures can often be described effectively by space-time trajectories of interacting particles, and their large scale behaviour depends on the overall geometry. We establish an exact relation between statistical properties of structures in uniformly expanding and fixed geometries, which depends only on the local scale invariance exponent and is independent of other properties such as the dimensionality. We illustrate our main result numerically in 1+1 dimensions for structures of coalescing L\'evy flights and fractional Brownian motions, including also branching, as well as for coalescencing finite size Brownian particles in 2+1 dimension. One of the main benefits is a full understanding of the asymptotic statistics in expanding domains, which are often non-trivial and random due to amplification of initial fluctuations.
Semiconductor quantum dots have a variety of applications in, e.g., quantum transport, qubit design, and solar-cell technology. In addition, they can be used as a computational playground to study complicated many-body quantum phenomena as well as chaotic effects. Here, the basics of quantum dots and their theoretical modeling are introduced. Then the attention is put to transport properties of stadium-shaped quantum dots that are shown to exhibit fractal conductance fluctuations in qualitative agreement with experiments. Finally, it is shown how quantum optimal control theory can be used to coherently control charge transfer and entanglement in quantum-dot systems.
The universal behaviour of 2D loop models can change dramatically when loops are allowed to cross. I will describe new phase transitions in such models and argue that they are driven by unbinding of point defects in an appropriate replica sigma model. I will use the field theory for the loop models to explain the phase diagram of a related model for polymer collapse, and will briefly describe a connection between the loop models and Anderson metal-insulator transitions.
The argument of the Evans Searles Fluctuation Theorem [1], namely the dissipation function [2] is also the key quantity in all linear and nonlinear response theory [3]. It is also the key quantity in the proof of the newly discovered equilibrium relaxation theorems. For the first time we have, subject to certain simple assumptions, a proof of thermal relaxation to the canonical distribution function [4] postulated by J. Willard Gibbs.
REFERENCES
[1] D.J. Evans and D.J. Searles, Phys. Rev. E 50,1645(1994).
[2] D.J. Searles and D.J. Evans, J.Chem. Phys., 113,3503(2000).
[3] D.J. Evans, D.J. Searles and S.R. Williams, J. Chem. Phys., 128, 014504(2008), ibid, 128, 249901(2008).
[4] D.J. Evans, D.J. Searles and S.R. Williams, J. Stat Mech.,P07029(2009).
Information theory provides a satisfactory theory for understanding stationary information sources. This theory, which was created to analyze communications between electronic devices, has found numerous applications in almost all branches of science.
A requirement to apply this theory is the existence of a fixed language, which is independent of the information that is shared. This makes this theory unsuitable for addressing fundamental questions of evolutionary biology, contemporary music cognition and many other disciplines. To the best of our knowledge, there exist no theory which is able to give account of evolving information sources and hence explain the dynamics of information.
There exist a deep link between information theory, which deals with stationary information sources, and equilibrium statistical mechanics. Thinking by analogy, we believe that non-equilibrium statistical mechanics holds the seeds for developing a theory which could explain the dynamics of information. The absence of the later may be related with the lack of a clear and general theory of non-equilibrium phenomena.
After reviewing the fundamentals concepts of information theory, the talk will present the limitations of the existent theory and explore the relationship between information dynamics and statistical physics.
The study of complex networks sheds light on the relation between the structure and function of complex systems. In this talk, we thoroughly discuss two paradigmatic examples of diffusion dynamics that take place on complex architectures: the spreading of diseases and the so-called complex social contagion. For the first process, we revisit the main theoretical and computational results obtained in the last decade. Specifically, we discuss several relevant cases ranging from the spreading of single diseases in single-layer networks, to the competitive dynamics of multi-strain diseases on top of metapopulation systems. Secondly, we discuss a different class of theoretical and numerical approaches suited to describe the emergence of collective phenomena in large-scale social systems. In particular, we show how this diffusion dynamics introduces different analytical and numerical challenges whose solution leads to a better understanding of the mechanisms at the root of the phenomena being analyzed.
Seminar series:
Statistical Mechanics Study Group
The main purpose of statistical mechanics is to give a microscopic derivation of macroscopic laws, including in particular the celebrated second law of thermodynamics. In recent years, there have been spectacular developments in this respect, including the integral and detailed work fluctuation theorems and the theory of stochastic thermodynamics. We give a brief introduction to these developments. In the first step, we derive the first and second law of thermodynamics for a Markovian stochastic process at the ensemble level, including two major advances: 1) the theory can be applied to small-scale systems including the effect of fluctuations, 2) the theory is not restricted to near-equilibrium dynamics. As an application, we evaluate the efficiency at maximum power of a two-state quan- tum dot. We also briefly discuss the connection to information-to-work conversion (Landauer principle). In a second step we formulate stochastic thermodynamics at the trajectory level, introducing stochastic trajectory-dependent quantities such as stochastic entropy, energy, heat, and work. Both the first and the second law can be formulated at this trajectory level. Concerning the second law, the crucial observation is that the stochastic entropy production can be written as the logarithm of the ratio of path probabilities. This in turn implies a detailed and integral work and fluctuation theorem, linking the probability to observe a given stochastic entropy production to that of observing minus this entropy change in a reverse experiment. The usual second law, stipulating the increase on average of the stochastic entropy production, follows as a subsidiary consequence.
The nonequilibrium classical dynamics and directed transport in lattices with
a spatially-dependent driving is explored. Prototype examples are phase, frequency
or amplitude-modulated lattices which, via a tuning of the parameters of the
driven unit cell, allow for an engineering of the classical phase space
and therefore of the magnitude and direction of the directed currents.
Several mechanisms for transient localization and trapping of particles
in different wells of the driven unit cell are presented and analyzed.
As a major first application we derive a mechanism for the patterned deposition
of particles in a spatio-temporally driven lattice. The working principle
is based on the breaking of the spatio-temporal translation symmetry, which is responsible
for the equivalence of all lattice sites. The patterned trapping of the particles occurs
in confined chaotic seas, created via the ramping of the height of the lattice potential.
Complex density profiles on the length scale of the complete lattice can be
obtained by a quasi-continuous, spatial deformation of the chaotic sea in a frequency modulated lattice.
In a second step we explore spatiotemporal superlattices consisting of
domains of differently time-driven spatial lattices. Here we demonstrate
a novel mechanisms for the conversion of ballistic to diffusive motion and vice versa.
This process takes place at the interfaces of domains subjected to different time-dependent
forces. As a consequence a complex short-time depletion dynamics at the interfaces followed
by long-time transient oscillations of the particle density are observed. The latter can
be converted to permanent density waves by an appropriate tuning
of the driving forces. The proposed mechanism opens the perspective of an
engineering of the nonequilibrium dynamics of particles in inhomogeneously driven lattices.
Finally we show the emergence of dynamical current reversals in long-range
interacting spatiotemporally driven lattices.
There are various cases of animal movement where behaviour broadly switches between two modes of operation, corresponding to a long distance movement state and a resting or local movement state. Here a mathematical description of this process is formulated, adapted from Friedrich et. al. (2006). The approach allows the specification any running or waiting time distribution along with any angular and speed distributions. The resulting system of partial integro-differential equations are tumultuous and therefore it is necessary to both simplify and derive summary statistics. An expression for the mean squared displacement is derived which shows good agreement with experimental data from the bacterium Escherichia coli and the gull Larus fuscus. Finally a large time diffusive approximation is considered via a Cattaneo approximation (Hillen, 2004). This leads to the novel result that the effective diffusion constant is dependent on the mean and variance of the running time distribution but only on the mean of the waiting time distribution.
I will talk about non-equilibrium quantum phenomena arising in the context of experimentally relevant condensed matter settings. Examples of the latter include quantum Hall (QHE) edge states driven out of equilibrium by applied bias voltage; and frustrated quantum magnets, where a sudden perturbation can result in an unusual non-equilibrium response. These systems show remarkable behaviour, such as emergence of non-equilibrium steady states, and a dynamical phase diagram, which arise as a result of fractionalization of electron/spin degrees of freedom into quasiparticles (e.g. Majorana fermions and fluxes of gauge field). I will present the theory of electron equilibration in QHE edge states, and discuss dynamical response in quantum spin-liquids.
[1] D.L. Kovrizhin and J.T. Chalker, Phys. Rev. Lett. 109, 106403 (2012)
[2] J. Knolle, D.L. Kovrizhin, J.T. Chalker, and R. Moessner, Phys. Rev. Lett. 112, 207203 (2014)
[1] E. Räsänen et al., PLoS ONE 10(6): e0127902 (2015).
[2] H. Hennig et al., Phys. Today. 65, 64 (2012).
carried out by Professor S Anlage in Maryland: A train of very short
electromagnetic pulses is fed to a network of coaxial transmission lines
through one vertex, and exit through another vertex. The times t it
takes to cross the network is the transit time of interest here. We
study the distribution of transit times and show that asymptotically it
falls off exponentially as Aexp(-ct) and explain how the constants A and
c depend on the network connectivity and the lengths of its edges.
See below for a pdf of Oscar's smartboard scribblings...
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See attachment for abstract.
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