arXiv subjects

Naftali R. Smith

Publications and source records attributed to Naftali R. Smith.

45 records · Page 3Linked to original sources

arXiv2019

Time-averaged height distribution of the Kardar-Parisi-Zhang interface

We study the complete probability distribution $\mathcal{P}\left(\bar{H},t\right)$ of the time-averaged height $\bar{H}=(1/t)\int_0^t h(x=0,t')\,dt'$ at point $x=0$ of an evolving 1+1 dimensional Kardar-Parisi-Zhang (KPZ) interface $h\left(x,t\right)$. We focus on short times and flat initial condition and employ the optimal fluctuation method to determine the variance and the third cumulant of the distribution, as well as the asymmetric stretched-exponential tails. The tails scale as $-\ln\mathcal{P}\sim\left|\bar{H}\right|^{3/2} \! /\sqrt{t}$ and $-\ln\mathcal{P}\sim\left|\bar{H}\right|^{5/2} \! /\sqrt{t}$, similarly to the previously determined tails of the one-point KPZ height statistics at specified time $t'=t$. The optimal interface histories, dominating these tails, are markedly different. Remarkably, the optimal history, $h\left(x=0,t\right)$, of the interface height at $x=0$ is a non-monotonic function of time: the maximum (or minimum) interface height is achieved at an intermediate time. We also address a more general problem of determining the probability density of observing a given height history of the KPZ interface at point $x=0$.

cond-mat.stat-mech↗

arXiv2019

A giant disparity and a dynamical phase transition in large deviations of the time-averaged size of stochastic populations

We study large deviations of the time-averaged size of stochastic populations described by a continuous-time Markov jump process. When the expected population size $N$ in the steady state is large, the large deviation function (LDF) of the time-averaged population size can be evaluated by using a WKB (after Wentzel, Kramers and Brillouin) method, applied directly to the master equation for the Markov process. For a class of models that we identify, the direct WKB method predicts a giant disparity between the probabilities of observing an unusually small and an unusually large values of the time-averaged population size. The disparity results from a qualitative change in the "optimal" trajectory of the underlying classical mechanics problem. The direct WKB method also predicts, in the limit of $N\to \infty$, a singularity of the LDF, which can be interpreted as a second-order dynamical phase transition. The transition is smoothed at finite $N$, but the giant disparity remains. The smoothing effect is captured by the van-Kampen system size expansion of the exact master equation near the attracting fixed point of the underlying deterministic model. We describe the giant disparity at finite $N$ by developing a different variant of WKB method, which is applied in conjunction with the Donsker-Varadhan large-deviation formalism and involves subleading-order calculations in $1/N$.

cond-mat.stat-mech↗

arXiv2019

Geometrical optics of constrained Brownian excursion: from the KPZ scaling to dynamical phase transitions

We study a Brownian excursion on the time interval $\left|t\right|\leq T$, conditioned to stay above a moving wall $x_{0}\left(t\right)$ such that $x_0\left(-T\right)=x_0\left(T\right)=0$, and $x_{0}\left(\left|t\right| 0$. For a whole class of moving walls, typical fluctuations of the conditioned Brownian excursion are described by the Ferrari-Spohn (FS) distribution and exhibit the Kardar-Parisi-Zhang (KPZ) dynamic scaling exponents $1/3$ and $2/3$. Here we use the optimal fluctuation method (OFM) to study atypical fluctuations, which turn out to be quite different. The OFM provides their simple description in terms of optimal paths, or rays, of the Brownian motion. We predict two singularities of the large deviation function, which can be interpreted as dynamical phase transitions, and they are typically of third order. Transitions of a fractional order can also appear depending on the behavior of $x_{0}\left(t\right)$ in a close vicinity of $t=\pm T$. Although the OFM does not describe typical fluctuations, it faithfully reproduces the near tail of the FS distribution and therefore captures the KPZ scaling. If the wall function $x_{0}\left(t\right)$ is not parabolic near its maximum, typical fluctuations (which we probe in the near tail) exhibit a more general scaling behavior with a continuous one-parameter family of scaling exponents.

cond-mat.stat-mech↗

arXiv2018

Comment on "Minimum Action Path Theory Reveals the Details of Stochastic Transitions Out of Oscillatory States"

De la Cruz et al. [Phys. Rev. Lett. 120, 128102 (2018); arXiv:1705.08683] studied a noise-induced transition in an oscillating stochastic population undergoing birth- and death-type reactions. They applied the Freidlin-Wentzell WKB formalism to determine the most probable path to the noise-induced escape from a limit cycle predicted by deterministic theory, and to find the probability distribution of escape time. Here we raise a number of objections to their calculations.

physics.bio-ph↗

arXiv2018

Exact short-time height distribution for the flat Kardar-Parisi-Zhang interface

We determine the exact short-time distribution $-\ln \mathcal{P}_{\text{f}}\left(H,t\right)= S_{\text{f}} \left(H\right)/\sqrt{t}$ of the one-point height $H=h(x=0,t)$ of an evolving 1+1 Kardar-Parisi-Zhang (KPZ) interface for flat initial condition. This is achieved by combining (i) the optimal fluctuation method, (ii) a time-reversal symmetry of the KPZ equation in 1+1 dimension, and (iii) the recently determined exact short-time height distribution $-\ln \mathcal{P}_{\text{st}}\left(H,t\right)= S_{\text{st}} \left(H\right)/\sqrt{t}$ for \emph{stationary} initial condition. In studying the large-deviation function $S_{\text{st}} \left(H\right)$ of the latter, one encounters two branches: an analytic and a non-analytic. The analytic branch is non-physical beyond a critical value of $H$ where a second-order dynamical phase transition occurs. Here we show that, remarkably, it is the analytic branch of $S_{\text{st}} \left(H\right)$ which determines the large-deviation function $S_{\text{f}} \left(H\right)$ of the flat interface via a simple mapping $S_{\text{f}}\left(H\right)=2^{-3/2}S_{\text{st}}\left(2H\right)$.

cond-mat.stat-mech↗

arXiv2018

Landau theory of the short-time dynamical phase transitions of the Kardar-Parisi-Zhang interface

We study the short-time distribution $\mathcal{P}\left(H,L,t\right)$ of the two-point two-time height difference $H=h(L,t)-h(0,0)$ of a stationary Kardar-Parisi-Zhang (KPZ) interface in 1+1 dimension. Employing the optimal-fluctuation method, we develop an effective Landau theory for the second-order dynamical phase transition found previously for $L=0$ at a critical value $H=H_c$. We show that $|H|$ and $L$ play the roles of inverse temperature and external magnetic field, respectively. In particular, we find a first-order dynamical phase transition when $L$ changes sign, at supercritical $H$. We also determine analytically $\mathcal{P}\left(H,L,t\right)$ in several limits away from the second-order transition. Typical fluctuations of $H$ are Gaussian, but the distribution tails are highly asymmetric. The tails $-\ln\mathcal{P}\sim\left|H\right|^{3/2} \! /\sqrt{t}$ and $-\ln\mathcal{P}\sim\left|H\right|^{5/2} \! /\sqrt{t}$, previously found for $L=0$, are enhanced for $L \ne 0$. At very large $|L|$ the whole height-difference distribution $\mathcal{P}\left(H,L,t\right)$ is time-independent and Gaussian in $H$, $-\ln\mathcal{P}\sim\left|H\right|^{2} \! /|L|$, describing the probability of creating a ramp-like height profile at $t=0$.

cond-mat.stat-mech↗

arXiv2018

Finite-size effects in the short-time height distribution of the Kardar-Parisi-Zhang equation

We use the optimal fluctuation method to evaluate the short-time probability distribution $\mathcal{P}\left(H,L,t\right)$ of height at a single point, $H=h\left(x=0,t\right)$, of the evolving Kardar-Parisi-Zhang (KPZ) interface $h\left(x,t\right)$ on a ring of length $2L$. The process starts from a flat interface. At short times typical (small) height fluctuations are unaffected by the KPZ nonlinearity and belong to the Edwards-Wilkinson universality class. The nonlinearity, however, strongly affects the (asymmetric) tails of $\mathcal{P}(H)$. At large $L/\sqrt{t}$ the faster-decaying tail has a double structure: it is $L$-independent, $-\ln\mathcal{P}\sim\left|H\right|^{5/2}/t^{1/2}$, at intermediately large $|H|$, and $L$-dependent, $-\ln\mathcal{P}\sim \left|H\right|^{2}L/t$, at very large $|H|$. The transition between these two regimes is sharp and, in the large $L/\sqrt{t}$ limit, behaves as a fractional-order phase transition. The transition point $H=H_{c}^{+}$ depends on $L/\sqrt{t}$. At small $L/\sqrt{t}$, the double structure of the faster tail disappears, and only the very large-$H$ tail, $-\ln\mathcal{P}\sim \left|H\right|^{2}L/t$, is observed. The slower-decaying tail does not show any $L$-dependence at large $L/\sqrt{t}$, where it coincides with the slower tail of the GOE Tracy-Widom distribution. At small $L/\sqrt{t}$ this tail also has a double structure. The transition between the two regimes occurs at a value of height $H=H_{c}^{-}$ which depends on $L/\sqrt{t}$. At $L/\sqrt{t} \to 0$ the transition behaves as a mean-field-like second-order phase transition. At $|H|<|H_c^{-}|$ the slower tail behaves as $-\ln\mathcal{P}\sim \left|H\right|^{2}L/t$, whereas at $|H|>|H_c^{-}|$ it coincides with the slower tail of the GOE Tracy-Widom distribution.

cond-mat.stat-mech↗

arXiv2016

Local average height distribution of fluctuating interfaces

Height fluctuations of growing surfaces can be characterized by the probability distribution of height in a spatial point at a finite time. Recently there has been spectacular progress in the studies of this quantity for the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimensions. Here we notice that, at or above a critical dimension, the finite-time one-point height distribution is ill-defined in a broad class of linear surface growth models, unless the model is regularized at small scales. The regularization via a system-dependent small-scale cutoff leads to a partial loss of universality. As a possible alternative, we introduce a \emph{local average height}. For the linear models the probability density of this quantity is well-defined in any dimension. The weak-noise theory (WNT) for these models yields the "optimal path" of the interface conditioned on a non-equilibrium fluctuation of the local average height. As an illustration, we consider the conserved Edwards-Wilkinson (EW) equation, where, without regularization, the finite-time one-point height distribution is ill-defined in all physical dimensions. We also determine the optimal path of the interface in a closely related problem of the finite-time \emph{height-difference} distribution for the non-conserved EW equation in $1+1$ dimension. Finally, we discuss a UV catastrophe in the finite-time one-point distribution of height in the (non-regularized) KPZ equation in $2+1$ dimensions.

cond-mat.stat-mech↗

arXiv2015

Extinction of oscillating populations

Established populations often exhibit oscillations in their sizes. If a population is isolated, intrinsic stochasticity of elemental processes can ultimately bring it to extinction. Here we study extinction of oscillating populations in a stochastic version of the Rosenzweig-MacArthur predator-prey model. To this end we extend a WKB approximation (after Wentzel, Kramers and Brillouin) of solving the master equation to the case of extinction from a limit cycle in the space of population sizes. We evaluate the extinction rates and find the most probable paths to extinction by applying Floquet theory to the dynamics of an effective WKB Hamiltonian. We show that the entropic barriers to extinction change in a non-analytic way as the system passes through the Hopf bifurcation. We also study the subleading pre-exponential factors of the WKB approximation.

cond-mat.stat-mech↗