Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem

Strong anomalous diffusion, where $\langle |x(t)|^q \rangle \sim t^{q \nu(q)}$ with a nonlinear spectrum \nu(q) \neq \mbox{const}, is wide spread and has been found in various nonlinear dynamical systems and experiments on active transport in living cells. Using a stochastic approach we show how this phenomena is related to infinite covariant densities, i.e., the asymptotic states of these systems are described by non-normalizable distribution functions. Our work shows that the concept of infinite covariant densities plays an important role in the statistical description of open systems exhibiting multi-fractal anomalous diffusion, as it is complementary to the central limit theorem.Comment: PRL, in pres

Universal fluctuations in subdiffusive transport

Subdiffusive transport in tilted washboard potentials is studied within the fractional Fokker-Planck equation approach, using the associated continuous time random walk (CTRW) framework. The scaled subvelocity is shown to obey a universal law, assuming the form of a stationary Levy-stable distribution. The latter is defined by the index of subdiffusion alpha and the mean subvelocity only, but interestingly depends neither on the bias strength nor on the specific form of the potential. These scaled, universal subvelocity fluctuations emerge due to the weak ergodicity breaking and are vanishing in the limit of normal diffusion. The results of the analytical heuristic theory are corroborated by Monte Carlo simulations of the underlying CTRW

Pressure in an exactly solvable model of active fluid

We consider the pressure in the steady-state regime of three stochastic models characterized by self-propulsion and persistent motion and widely employed to describe the behavior of active particles, namely the Active Brownian particle (ABP) model, the Gaussian colored noise (GCN) model and the unified colored noise model (UCNA). Whereas in the limit of short but finite persistence time the pressure in the UCNA model can be obtained by different methods which have an analog in equilibrium systems, in the remaining two models only the virial route is, in general, possible. According to this method, notwithstanding each model obeys its own specific microscopic law of evolution, the pressure displays a certain universal behavior. For generic interparticle and confining potentials, we derive a formula which establishes a correspondence between the GCN and the UCNA pressures. In order to provide explicit formulas and examples, we specialize the discussion to the case of an assembly of elastic dumbbells confined to a parabolic well. By employing the UCNA we find that, for this model, the pressure determined by the thermodynamic method coincides with the pressures obtained by the virial and mechanical methods. The three methods when applied to the GCN give a pressure identical to that obtained via the UCNA. Finally, we find that the ABP virial pressure exactly agrees with the UCNA and GCN result.Comment: 12 pages, 1 figure Submitted for publication 23rd of January 2017 The introduction has been modifie

Time evolution towards q-Gaussian stationary states through unified Ito-Stratonovich stochastic equation

We consider a class of single-particle one-dimensional stochastic equations which include external field, additive and multiplicative noises. We use a parameter $\theta \in [0,1]$ which enables the unification of the traditional It\^o and Stratonovich approaches, now recovered respectively as the $\theta=0$ and $\theta=1/2$ particular cases to derive the associated Fokker-Planck equation (FPE). These FPE is a {\it linear} one, and its stationary state is given by a $q$-Gaussian distribution with $q = \frac{\tau + 2M (2 - \theta)}{\tau + 2M (1 - \theta)}<3$, where $\tau \ge 0$ characterizes the strength of the confining external field, and $M \ge 0$ is the (normalized) amplitude of the multiplicative noise. We also calculate the standard kurtosis $\kappa_1$ and the $q$-generalized kurtosis $\kappa_q$ (i.e., the standard kurtosis but using the escort distribution instead of the direct one). Through these two quantities we numerically follow the time evolution of the distributions. Finally, we exhibit how these quantities can be used as convenient calibrations for determining the index $q$ from numerical data obtained through experiments, observations or numerical computations.Comment: 9 pages, 2 figure

Scaling Relations and Exponents in the Growth of Rough Interfaces Through Random Media

The growth of a rough interface through a random media is modelled by a continuous stochastic equation with a quenched noise. By use of the Novikov theorem we can transform the dependence of the noise on the interface height into an effective temporal correlation for different regimes of the evolution of the interface. The exponents characterizing the roughness of the interface can thus be computed by simple scaling arguments showing a good agreement with recent experiments and numerical simulations.Comment: 4 pages, RevTex, twocolumns, two figures (upon request). To appear in Europhysics Letter

Viscosity Dependence of the Folding Rates of Proteins

The viscosity dependence of the folding rates for four sequences (the native state of three sequences is a beta-sheet, while the fourth forms an alpha-helix) is calculated for off-lattice models of proteins. Assuming that the dynamics is given by the Langevin equation we show that the folding rates increase linearly at low viscosities \eta, decrease as 1/\eta at large \eta and have a maximum at intermediate values. The Kramers theory of barrier crossing provides a quantitative fit of the numerical results. By mapping the simulation results to real proteins we estimate that for optimized sequences the time scale for forming a four turn \alpha-helix topology is about 500 nanoseconds, whereas the time scale for forming a beta-sheet topology is about 10 microseconds.Comment: 14 pages, Latex, 3 figures. One figure is also available at http://www.glue.umd.edu/~klimov/seq_I_H.html, to be published in Physical Review Letter

Josephson-like currents in graphene for arbitrary time-dependent potential barriers

From the exact solution of the Dirac-Weyl equation we find unusual currents j_y running in y-direction parallel to a time-dependent scalar potential barrier W(x,t) placed upon a monolayer of graphene, even for vanishing momentum component p_y. In their sine-like dependence on the phase difference of wave functions, describing left and right moving Dirac fermions, these currents resemble Josephson currents in superconductors, including the occurance of Shapiro steps at certain frequencies of potential oscillations. The Josephson-like currents are calculated for several specific time-dependent barriers. A novel type of resonance is discovered when, accounting for the Fermi velocity, temporal and spatial frequencies match.Comment: 8 pages, 4 figur

Polarons as Nucleation Droplets in Non-Degenerate Polymers

We present a study of the nucleation mechanism that allows the decay of the metastable phase (trans-cisoid) to the stable phase (cis-transoid) in quasi one-dimensional non-degenerate polymers within the continuum electron-phonon model. The electron-phonon configurations that lead to the decay, i.e. the critical droplets (or transition state), are identified as polarons of the metastable phase. We obtain an estimate for the decay rate via thermal activation within a range of parameters consistent with experimental values for the gap of the cis-configuration. It is pointed out that, upon doping, the activation barriers of the excited states are quite smaller and the decay rate is greatly enhanced. Typical activation energies for electron or hole polarons are $\approx 0.1$ eV and the typical size for a critical droplet (polaron) is about $20 \AA$. Decay via quantum nucleation is also studied and it is found that the crossover temperature between quantum nucleation and thermal activation is of order $T_c \leq 40 ^oK$. Metastable configurations of non-degenerate polymers may provide examples for mesoscopic quantum tunneling.Comment: REVTEX 3.0, 28 PAGES, 3 FIGURES AVAILABLE UPON REQUEST, PITT 94-0

Real Time Analysis of Thermal Activation via Sphaleron Transitions

We study the process of thermal activation mediated by sphaleron transitions by analyzing the real-time dynamics of the decay out of equilibrium in a $1+1$ dimensional field theory with a metastable state. The situation considered is that of a rapid supercooling in which the system is trapped in a metastable state at a temperature larger than the mass of the quanta, but smaller than the energy to create a critical droplet. The initial density matrix is evolved in time and the nucleation rate (probability current at the saddle point) is computed. The nucleation rate is {\it time dependent}, vanishing at early times, reaching a maximum at a time $t \approx 1/m$ with $m$ the mass of quanta in the metastable state, and decreasing at long times as a consequence of unitarity. An estimate for the average number of particles of ``true vacuum'' produced as a function of time during the nucleation process is obtained.Comment: 26 pages 6 figures (available upon request), PITT-93-06, LATE

Work and heat fluctuations in two-state systems: a trajectory thermodynamics formalism

Two-state models provide phenomenological descriptions of many different systems, ranging from physics to chemistry and biology. We investigate work fluctuations in an ensemble of two-state systems driven out of equilibrium under the action of an external perturbation. We calculate the probability density P(W) that a work equal to W is exerted upon the system along a given non-equilibrium trajectory and introduce a trajectory thermodynamics formalism to quantify work fluctuations in the large-size limit. We then define a trajectory entropy S(W) that counts the number of non-equilibrium trajectories P(W)=exp(S(W)/kT) with work equal to W. A trajectory free-energy F(W) can also be defined, which has a minimum at a value of the work that has to be efficiently sampled to quantitatively test the Jarzynski equality. Within this formalism a Lagrange multiplier is also introduced, the inverse of which plays the role of a trajectory temperature. Our solution for P(W) exactly satisfies the fluctuation theorem by Crooks and allows us to investigate heat-fluctuations for a protocol that is invariant under time reversal. The heat distribution is then characterized by a Gaussian component (describing small and frequent heat exchange events) and exponential tails (describing the statistics of large deviations and rare events). For the latter, the width of the exponential tails is related to the aforementioned trajectory temperature. Finite-size effects to the large-N theory and the recovery of work distributions for finite N are also discussed. Finally, we pay particular attention to the case of magnetic nanoparticle systems under the action of a magnetic field H where work and heat fluctuations are predicted to be observable in ramping experiments in micro-SQUIDs.Comment: 28 pages, 14 figures (Latex