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Register a new userStatistical mechanics of a single active slider on a fluctuating interface
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We study the statistical mechanics of a single active slider on a fluctuating interface, by means of numerical simulations and theoretical arguments. The slider, which moves by definition towards the interface minima, is active as it also stimulates growth of the interface. Even though such a particle has no counterpart in thermodynamic systems, active sliders may provide a simple model for ATP-dependent membrane proteins that activate cytoskeletal growth. We find a wide range of dynamical regimes according to the ratio between the timescales associated with the slider motion and the interface relaxation. If the interface dynamics is slow, the slider behaves like a random walker in a random envinronment which, furthermore, is able to escape environmental troughs by making them grow. This results in different dynamic exponens to the interface and the particle: the former behaves as an Edward-Wilkinson surface with dynamic exponent 2 whereas the latter has dynamic exponent 3/2. When the interface is fast, we get sustained ballistic motion with the particle surfing a membrane wave created by itself. However, if the interface relaxes immediately (i.e., it is infinitely fast), particle motion becomes symmetric and goes back to diffusive. Due to such a rich phenomenology, we propose the active slider as a toy model of fundamental interest in the field of active membranes and, generally, whenever the system constituent can alter the environment by spending energy.
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We study pattern formation, fluctuations and scaling induced by a growth-promoting active walker on an otherwise static interface. Active particles on an interface define a simple model for energy consuming proteins embedded in the plasma membrane, responsible for membrane deformation and cell movement. In our model, the active particle overturns local valleys of the interface into hills, simulating growth, while itself sliding and seeking new valleys. In 1D, this overturn-slide-search dynamics of the active particle causes it to move superdiffusively in the transverse direction while pulling the immobile interface upwards. Using Monte Carlo simulations, we find an emerging tent-like mean profile developing with time, despite large fluctuations. The roughness of the interface follows scaling with the growth, dynamic and roughness exponents, derived using simple arguments as $beta=2/3, z=3/2, alpha=1/2$ respectively, implying a breakdown of the usual scaling law $beta = alpha/z$, owing to very local growth of the interface. The transverse displacement of the puller on the interface scales as $sim t^{2/3}$ and the probability distribution of its displacement is bimodal, with an unusual linear cusp at the origin. Both the mean interface pattern and probability distribution display scaling. A puller on a static 2D interface also displays aspects of scaling in the mean profile and probability distribution. We also show that a pusher on a fluctuating interface moves subdiffusively leading to a separation of time scale in pusher motion and interface response.
These lectures were prepared for the 2014 PCMI graduate summer school and were designed to be a lightweight introduction to statistical mechanics for mathematicians. The applications feature some of the themes of the summer school: sphere packings and tilings.
We review the field of the glass transition, glassy dynamics and aging from a statistical mechanics perspective. We give a brief introduction to the subject and explain the main phenomenology encountered in glassy systems, with a particular emphasis on spatially heterogeneous dynamics. We review the main theoretical approaches currently available to account for these glassy phenomena, including recent developments regarding mean-field theory of liquids and glasses, novel computational tools, and connections to the jamming transition. Finally, the physics of aging and off-equilibrium dynamics exhibited by glassy materials is discussed.
For a given thermodynamic system, and a given choice of coarse-grained state variables, the knowledge of a force-flux constitutive law is the basis for any nonequilibrium modeling. In the first paper of this series we established how, by a generalization of the classical fluctuation-dissipation theorem (FDT), the structure of a constitutive law is directly related to the distribution of the fluctuations of the state variables. When these fluctuations can be expressed in terms of diffusion processes, one may use Green-Kubo-type coarse-graining schemes to find the constitutive laws. In this paper we propose a coarse-graining method that is valid when the fluctuations are described by means of general Markov processes, which include diffusions as a special case. We prove the success of the method by numerically computing the constitutive law for a simple chemical reaction $A rightleftarrows B$. Furthermore, we show that one cannot find a consistent constitutive law by any Green-Kubo-like scheme.
This work describes a simple agent model for the spread of an epidemic outburst, with special emphasis on mobility and geographical considerations, which we characterize via statistical mechanics and numerical simulations. As the mobility is decreased, a percolation phase transition is found separating a free-propagation phase in which the outburst spreads without finding spatial barriers and a localized phase in which the outburst dies off. Interestingly, the number of infected agents is subject to maximal fluctuations at the transition point, building upon the unpredictability of the evolution of an epidemic outburst. Our model also lends itself to test with vaccination schedules. Indeed, it has been suggested that if a vaccine is available but scarce it is convenient to select carefully the vaccination program to maximize the chances of halting the outburst. We discuss and evaluate several schemes, with special interest on how the percolation transition point can be shifted, allowing for higher mobility without epidemiological impact.
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