A circadian oscillator driven by external noise and internal noise has been studied by use of the chemical Langevin equation method. When the system is near a Hopf bifurcation and driven by internal noise only, it is found that the coherence resonance phenomenon can be induced by the internal noise. When the system is simultaneously driven by internal and external noise, it is found that external-noise coherence resonance can be suppressed by internal noise, while internal-noise coherence resonance can be enhanced by modulation of the external noise intensity in a certain range of noise intensity. Another interesting result is that the external noise can regulate the optimal system size when the internal-noise coherence resonance occurs.

An analytic description for traveling waves in a one-dimensional reaction-diffusion system with one activator and two inhibitors and with equal diffusion constants is developed using a piecewise linear approximation for the nonlinear activator reaction term. The case of front waves is examined in more detail, the monotonic and oscillating fronts being separately considered. The corresponding wave profiles are constructed, and the speed equation is obtained and discussed. It is found that the fronts in the three-component model propagate faster than the fronts in the two-component system. The front interaction is studied using numerical calculations. The results show that at head-on collisions two oscillating fronts produce a wavy domain, which spreads in space with time.

The stability of a planar impulse in rectangular spatial domains for a two-variable excitable reaction-diffusion system is numerically studied. The dependence of the stability on the size of the domain perpendicular to the direction of the propagation of the impulse is shown. The instability results in asymptotic stable curved impulses or an asymptotic spatiotemporal structure, which is generated similarly to the one-dimensional backfiring phenomenon.

A quantum fluctuation theorem for a driven quantum subsystem interacting with its environment is derived based solely on the assumption that its reduced density matrix obeys a closed evolution equation—i.e., a quantum master equation (QME). Quantum trajectories and their associated entropy, heat, and work appear naturally by transforming the QME to a time-dependent Liouville space basis that diagonalizes the instantaneous reduced density matrix of the subsystem. A quantum integral fluctuation theorem, a steady-state fluctuation theorem, and the Jarzynski relation are derived in a similar way as for classical stochastic dynamics.

We compute exact analytical expressions for the skewness and kurtosis in the two-dimensional harmonic $XY$ model. These quantities correspond to the third and fourth normalized moments of the probability density function (PDF) of the magnetization of the model. From their behavior, we conclude that they depend explicitly on the system temperature even in the thermodynamic limit, and hence the PDF itself must depend on it. Our results correct the hypothesis called universal fluctuations, they confirm and extend previous results which showed a $T$ dependence of the PDF, including perturbative expansions within the $XY$ model up to first order in temperature.

We investigate a $(1+1)$-dimensional nonlinear field theoretic model with the field potential $V\left(\varphi \right)=\mid \varphi \mid$. It can be obtained as the universal small amplitude limit in a class of models with potentials which are symmetrically V-shaped at their minima, or as a continuum limit of certain mechanical system with an infinite number of degrees of freedom. We point out that the model has an interesting scaling symmetry. One consequence of that is the existence of self-similar solutions of the corresponding field equations. We also find discontinuous solutions of shock wave type.

We study the critical relaxation of the helix-coil transition in all-atom models of polyalanine chains. We show that at the critical temperature the decay of a completely helical conformation can be described by scaling relations that allow us estimating the pertinent critical exponents. The present approach opens a new way for characterizing transitions in proteins and may lead to a better understanding of their folding mechanism. An application of the technique to the 34-residue human parathyroid fragment PTH(1-34) supports universality of the helix-coil transition in homopolymers and (helical) proteins.

A procedure suggested by Vvedensky for obtaining continuum equations as the coarse-grained limit of discrete models is applied to the restricted solid-on-solid model with both adsorption and desorption. Using an expansion of the master equation, discrete Langevin equations are derived; these agree quantitatively with direct simulation of the model. From these, a continuum differential equation is derived, and the model is found to exhibit either Edwards-Wilkinson or Kardar-Parisi-Zhang exponents, as expected from symmetry arguments. The coefficients of the resulting continuum equation remain well-defined in the coarse-grained limit.

Problems in nonequilibrium statistical physics are characterized by the absence of a fluctuation dissipation theorem. The usual analytic route for treating these vast class of problems is to use response fields in addition to the real fields that are pertinent to a given problem. This line of argument was introduced by Martin, Siggia, and Rose. We show that instead of using the response field, one can, following the stochastic quantization of Parisi and Wu, introduce a fictitious time. In this extra dimension a fluctuation dissipation theorem is built in and provides a different outlook to problems in nonequilibrium statistical physics.

We explore stochastic models for the study of ion transport in biological cells. Analysis of these models explains and explores an interesting feature of ion transport observed by biophysicists. Namely, the average time it takes ions to cross certain ion channels is the same in either direction, even if there is an electric potential difference across the channels. It is shown for simple single ion models that the distribution of a path (i.e., the history of location versus time) of an ion crossing the channel in one direction has the same distribution as the time-reversed path of an ion crossing the channel in the reverse direction. Therefore, not only is the mean duration of these paths equal, but other measures, such as the variance of passage time or the mean time a path spends within a specified section of the channel, are also the same for both directions of traversal. The feature is also explored for channels with interacting ions. If a system of interacting ions is in reversible equilibrium (net flux is zero), then the equivalence of the left-to-right trans paths with the time-reversed right-to-left trans paths still holds. However, if the system is in equilibrium, but not reversible equilibrium, then such equivalence need not hold.

We develop variational principles to study the structure and the stability of equilibrium states of axisymmetric flows. We show that the axisymmetric Euler equations for inviscid flows admit an infinite number of steady state solutions. We find their general form and provide analytical solutions in some special cases. The system can be trapped in one of these steady states as a result of an inviscid violent relaxation. We show that the stable steady states maximize a (nonuniversal) $H$ function while conserving energy, helicity, circulation, and angular momentum (robust constraints). This can be viewed as a form of generalized selective decay principle. We derive relaxation equations which can be used as numerical algorithm to construct nonlinearly dynamically stable stationary solutions of axisymmetric flows. We also develop a thermodynamical approach to predict the equilibrium state at some fixed coarse-grained scale. We show that the resulting distribution can be divided in two parts: one universal coming from the conservation of robust invariants and one non-universal determined by the initial conditions through the fragile invariants (for freely evolving systems) or by a prior distribution encoding nonideal effects such as viscosity, small-scale forcing, and dissipation (for forced systems). Finally, we derive a parametrization of inviscid mixing to describe the dynamics of the system at the coarse-grained scale. A conceptual interest of this axisymmetric model is to be intermediate between two-dimensional (2D) and 3D turbulence.

Renormalization group calculations for fluctuation-dominated reaction-diffusion systems are generally in agreement with simulations and exact solutions. However, simulations of the single-species reactions $3A\to (\stackrel{̸}{0},A,2A)$ at their upper critical dimension $d_c=1$ have found asymptotic densities argued to be inconsistent with renormalization group predictions. We show that this discrepancy is resolved by inclusion of the leading corrections to scaling, which we derive explicitly and show to be universal, a property not shared by the $A+A\to (\stackrel{̸}{0},A)$ reactions. Finally, we demonstrate that two previous Smoluchowski approaches to this problem reduce, with various corrections, to a single theory which surprisingly yields the same asymptotic density as the renormalization group.

An efficient technique for conducting rotating tumbler experiments in parallel is introduced and used to study the effect of tumbler length and periodic tilting of the tumbler on axial segregation. When rotated, bidisperse granular slurries segregate into what appear at the surface to be alternating bands of larger and smaller particles. The number of bands increases linearly with tumbler length while the fractional area occupied by each type of band is constant. Periodic tilting of the rotation axis induces a periodic axial flow of particles in the flowing layer. For the range of tilt angle amplitudes investigated $(0^{\circ }\text{–}3.5^{\circ })$, the number of bands decreases with increasing angle, but the rate of merging and the fractional area of bands rich in smaller particles are unaffected.

Formation of microdomain structures in concentrated systems of irregular (stochastic) copolymers with weak composition asymmetry $(ϵ⪡1)$ is considered theoretically. The study is focused on the weak segregation regime for infinitely long copolymer chains near the disorder-to-order transition. It is shown that the transition occurs below the spinodal (at ${\chi }_0<{\chi }^{*}$) and that it results in the formation of micelles rather than in a superposition pattern of a few harmonic composition waves. The micelle size is inversely proportional to $\Delta \chi =\chi -{\chi }_0$. For small $\Delta \chi$ the micelles are like large nearly uniform droplets with “reflected” composition, $-ϵ$. As $\Delta \chi$ increases the micelle composition profile develops oscillations. A first-order transition from spherical-wave micelles to micelles with internal bcc structure is predicted at $\chi ∕{\chi }^{*}-1\propto {ϵ}^2$. Interaction of micelles is also considered. It is shown that micelles always tend to form a fcc superlattice. The micellization in the weak segregation regime is controlled by the so-called nonlocal free energy. The classical fourth order (in composition parameter $A$) approximation for this energy is significantly generalized in the regime of low volume fraction of micelles. It is shown that the nonlocal energy strongly (exponentially) increases with $A$ near and above a certain critical value $A\simeq A^{*}$.

The dynamic properties of the regulatory network governing the choice between lytic and lysogenic growths of coliphage lambda is studied using a Markov chain stochastic model. Our computer simulation confirms the finding by Li et al. [Proc. Natl. Acad. Sci. USA 101, 4781 (2004)] on the dynamics of budding yeast: that the biological stationary states are global attractors; the biological pathways of lytic and lysogenic growths are attracting trajectories; and the network functions are robustly designed against structural perturbations. In addition, our model shows the stochastic switch from lysogen to lytic growth, which has been observed in experiments. A definition of pseudoenergy is introduced in the network dynamics to reveal a transitionlike behavior in the system.

Here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagomé and $(3,{12}^2)$ lattices. We present two different methods: one which provides an approximation to the inhomogeneous kagomé and $(3,{12}^2)$ bond problems, and the other which gives estimates of $p_c$ for the homogeneous kagomé (0.524 408 8…) and $(3,{12}^2)$ (0.740 421 2…) problems that, respectively, agree with numerical results to five and six significant figures.

We consider the modified Ising model introduced by de Oliveira, Mendes, and Santos [J. Phys. A 26, 2317 (1993)], where the temperature depends locally on the spin configuration and detailed balance and local equilibrium are not obeyed. We derive a relation between the linear response function and correlation functions that generalizes the fluctuation-dissipation theorem. In the stationary states of the model, which are the counterparts of the Ising equilibrium states, the fluctuation-dissipation theorem breaks down due to the lack of time reversal invariance. In the nonstationary phase-ordering kinetics, the parametric plot of the integrated response function $\chi (t,t_w)$ vs the autocorrelation function is different from that of the kinetic Ising model. However, splitting $\chi (t,t_w)$ into a stationary and an aging term $\chi (t,t_w)={\chi }_{\mathit{st}}(t-t_w)+{\chi }_{\mathit{ag}}(t,t_w)$, we find ${\chi }_{\mathit{ag}}(t,t_w)\sim t_w^{-a_{\chi }}f(t∕t_w)$, and a numerical value of $a_{\chi }$ consistent with $a_{\chi }=\frac{1}{4}$, as in the kinetic Ising model.

We investigate the classical and quantum dynamics of the open quartic oscillator model. Typically quantum behavior such as collapses and revivals (also squeezing) are induced by the nonlinearity of the model. We show that purely diffusive environments, as expected, attenuate such phenomena. We obtain analytical results in both regimes classical and quantum and discuss the effect of a diffusive reservoir in the two cases. We show that “separation times” as usually defined in the literature are strongly observable (and initial condition) dependent, rendering a solid definition of a unique classical limit rather difficult. In particular, the separation time for the variance $⟨\mathrm{\Delta }\widehat{x}⟩$ can be smaller than that for the expectation value of the position $⟨\widehat{x}⟩$ of the centroid of the wave packet. We find a hierarchy of time scales which depends on the observable and the reservoir.

Interacting many-particle systems with a mean-field one-body part plus a chaos generating random two-body interaction having strength $\lambda$ exhibit Poisson to Gaussian orthogonal ensemble and Breit-Wigner (BW) to Gaussian transitions in level fluctuations and strength functions with transition points marked by $\lambda ={\lambda }_c$ and $\lambda ={\lambda }_F$, respectively; ${\lambda }_F⪢{\lambda }_c$. For these systems a theory for the matrix elements of one-body transition operators is available, as valid in the Gaussian domain, with $\lambda >{\lambda }_F$, in terms of orbital occupation numbers, level densities, and an integral involving a bivariate Gaussian in the initial and final energies. Here we show that, using a bivariate-$t$ distribution, the theory extends below from the Gaussian regime to the BW regime up to $\lambda ={\lambda }_c$. This is well tested in numerical calculations for 6 spinless fermions in 12 single-particle states.