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Artificial neural networks often struggle with catastrophic forgetting when learning multiple tasks sequentially, as training on new tasks degrades the performance on previously learned ones. Recent theoretical work has addressed this issue by analysing learning curves in synthetic frameworks under predefined training protocols. However, these protocols relied on heuristics and lacked a solid theoretical foundation assessing their optimality. In this paper, we fill this gap combining exact equations for training dynamics, derived using statistical physics techniques, with optimal control methods. We apply this approach to teacher-student models for continual learning and multi-task problems, obtaining a theory for task-selection protocols maximising performance while minimising forgetting. Our theoretical analysis offers non-trivial yet interpretable strategies for mitigating catastrophic forgetting, shedding light on how optimal learning protocols can modulate established effects, such as the influence of task similarity on forgetting. Finally, we validate our theoretical findings on real-world data., Comment: 19 pages, 9 figures

We consider a discrete-time random walk on a line starting at $x_0\geq 0$ where a cost is incurred at each jump. We obtain an exact analytical formula for the distribution of the total cost of a trajectory until the process crosses the origin for the first time. The formula is valid for arbitrary jump distribution and cost function (heavy- and light-tailed alike), provided they are symmetric and continuous. We analyze the formula in different scaling regimes, and find a high degree of universality with respect to the details of the jump distribution and the cost function. Applications are given to the motion of an active run-and-tumble particle in one dimension and extensions to multiple cost variables are considered. The analytical results are in perfect agreement with numerical simulations., Comment: 27 pages, 9 figures. Extended version which includes a detailed analysis of the $x_0>0$ case

Accretion flows are fundamentally turbulent systems, yet are classically modelled with viscous theories only valid on length scales significantly greater than the typical size of turbulent eddies in the flow. We demonstrate that, while this will be a reasonable bulk description of the flow at large radii, this must break down as the flow approaches absorbing boundaries, such as the innermost stable circular orbit (ISCO) of a black hole disc. This is because in a turbulent flow large velocity fluctuations can carry a fluid element over the ISCO from a finite distance away, from which it will not return, a process without analogy in conventional models. This introduces a non-zero directional bias into the velocity fluctuations in the near-ISCO disc. By studying reduced random walk problems, we derive a number of implications of the presence of an absorbing boundary in an accretion context. In particular, we show that the average velocity with which a typical fluid element crosses the ISCO is much larger than is assumed in traditional theories. This enhanced velocity modifies the thermodynamic properties of black hole accretion flows on both sides of the ISCO. In particular, thermodynamic quantities for larger ISCO stresses no longer display pronounced cusps at the ISCO in this new formalism, a result with relevance for a number of observational probes of the intra-ISCO region. Finally, we demonstrate that these extended models reproduce the trans-ISCO behaviour observed in GRMHD simulations of thin discs., Comment: 14 pages + appendices. 10 figures. Resubmitted to MNRAS

Recent experiments have implemented resetting by means of an external trap, whereby a system relaxes to the minimum of the trap and is reset in a finite time. In this work, we set up and analyse the thermodynamics of such a protocol. We present a general framework, even valid for non-Poissonian resetting, that captures the thermodynamic work required to maintain a resetting process up to a given observation time, and exactly calculate the moment generating function of this work. Our framework is valid for a wide range of systems, the only assumption being relaxation to equilibrium in the resetting trap. Examples and extensions are considered. In the case of Brownian motion, we investigate optimal resetting schemes that minimise work and its fluctuations, the mean work for arbitrary switching protocols and comparisons to previously studied resetting schemes. Numerical simulations are performed to validate our findings.

We consider the Nonlinear-Cost Random Walk model in discrete time introduced in [Phys. Rev. Lett. 130, 237102 (2023)], where a fee is charged for each jump of the walker. The nonlinear cost function is such that slow/short jumps incur a flat fee, while for fast/long jumps the cost is proportional to the distance covered. In this paper we compute analytically the average and variance of the distance covered in $n$ steps when the total budget $C$ is fixed, as well as the statistics of the number of long/short jumps in a trajectory of length $n$, for the exponential jump distribution. These observables exhibit a very rich and non-monotonic scaling behavior as a function of the variable $C/n$, which is traced back to the makeup of a typical trajectory in terms of long/short jumps, and the resulting "entropy" thereof. As a byproduct, we compute the asymptotic behavior of ratios of Kummer hypergeometric functions when both the first and last arguments are large. All our analytical results are corroborated by numerical simulations., Comment: 31 pages, 8 figures

We use linear stability analysis and hybrid lattice Boltzmann simulations to study the dynamical behaviour of an active nematic confined in a channel made of viscoelastic material. We find that the quiescent, ordered active nematic is unstable above a critical activity. The transition is to a steady flow state for high elasticity of the channel surroundings. However, below a threshold elastic modulus, the system produces spontaneous oscillations with periodic flow reversals. We provide a phase diagram that highlights the region where time-periodic oscillations are observed and explain how they are produced by the interplay of activity and viscoelasticity. Our results suggest new experiments to study the role of viscoelastic confinement in the spatio-temporal organization and control of active matter., Comment: Main text: 5 pages + 4 figs., Supp. Mat.: 4 pages

We consider a system of non-interacting particles on a line with initial positions distributed uniformly with density $\rho$ on the negative half-line. We consider two different models: (i) each particle performs independent Brownian motion with stochastic resetting to its initial position with rate $r$ and (ii) each particle performs run and tumble motion, and with rate $r$ its position gets reset to its initial value and simultaneously its velocity gets randomised. We study the effects of resetting on the distribution $P(Q,t)$ of the integrated particle current $Q$ up to time $t$ through the origin (from left to right). We study both the annealed and the quenched current distributions and in both cases, we find that resetting induces a stationary limiting distribution of the current at long times. However, we show that the approach to the stationary state of the current distribution in the annealed and the quenched cases are drastically different for both models. In the annealed case, the whole distribution $P_{\rm an}(Q,t)$ approaches its stationary limit uniformly for all $Q$. In contrast, the quenched distribution $P_{\rm qu}(Q,t)$ attains its stationary form for $Q Q_{\rm crit}(t)$. We show that $Q_{\rm crit}(t)$ increases linearly with $t$ for large $t$. On the scale where $Q \sim Q_{\rm crit}(t)$, we show that $P_{\rm qu}(Q,t)$ has an unusual large deviation form with a rate function that has a third-order phase transition at the critical point. We have computed the associated rate functions analytically for both models. Using an importance sampling method that allows to probe probabilities as tiny as $10^{-14000}$, we were able to compute numerically this non-analytic rate function for the resetting Brownian dynamics and found excellent agreement with our analytical prediction., Comment: 26 pages, 6 figures

We introduce a simple model of diffusive jump process where a fee is charged for each jump. The nonlinear cost function is such that slow jumps incur a flat fee, while for fast jumps the cost is proportional to the velocity of the jump. The model -- inspired by the way taxi meters work -- exhibits a very rich behavior. The cost for trajectories of equal length and equal duration exhibits giant fluctuations at a critical value of the scaled distance travelled. Furthermore, the full distribution of the cost until the target is reached exhibits an interesting ``freezing'' transition in the large-deviation regime. All the analytical results are corroborated by numerical simulations. Our results also apply to elastic systems near the depinning transition, when driven by a random force., Comment: 4 fig. 6 pages + Supplemental Material included. Published version, which includes physical applications to elastic systems in presence of random forces. Title changed to match published version

Stochastic systems that undergo random restarts to their initial state have been widely investigated in recent years, both theoretically and in experiments. Oftentimes, however, resetting to a fixed state is impossible due to thermal noise or other limitations. As a result, the system configuration after a resetting event is random. Here, we consider such a resetting protocol for an overdamped Brownian particle in a confining potential $V(x)$. We assume that the position of the particle is reset at a constant rate to a random location $x$, drawn from a distribution $p_R(x)$. To investigate the thermodynamic cost of resetting, we study the stochastic entropy production $S_{\rm Total}$. We derive a general expression for the average entropy production for any $V(x)$, and the full distribution $P(S_{\rm Total}|t)$ of the entropy production for $V(x)=0$. At late times, we show that this distribution assumes the large-deviation form $P(S_{\rm Total}|t)\sim \exp\left[-t^{2\alpha-1}\phi\left(\left(S_{\rm Total}-\langle S_{\rm Total}\rangle\right)/t^{\alpha}\right)\right]$, with $1/2<\alpha\leq 1$. We compute the rate function $\phi(z)$ and the exponent $\alpha$ for exponential and Gaussian resetting distributions. In the latter case, we find the anomalous exponent $\alpha=2/3$ and show that $\phi(z)$ has a first-order singularity at a critical value of $z$, corresponding to a real-space condensation transition., Comment: 29 pages, 6 figures

We consider a one-dimensional stationary time series of fixed duration $T$. We investigate the time $t_{\rm m}$ at which the process reaches the global maximum within the time interval $[0,T]$. By using a path-decomposition technique, we compute the probability density function $P(t_{\rm m}|T)$ of $t_{\rm m}$ for several processes, that are either at equilibrium (such as the Ornstein-Uhlenbeck process) or out of equilibrium (such as Brownian motion with stochastic resetting). We show that for equilibrium processes the distribution of $P(t_{\rm m}|T)$ is always symmetric around the midpoint $t_{\rm m}=T/2$, as a consequence of the time-reversal symmetry. This property can be used to detect nonequilibrium fluctuations in stationary time series. Moreover, for a diffusive particle in a confining potential, we show that the scaled distribution $P(t_{\rm m}|T)$ becomes universal, i.e., independent of the details of the potential, at late times. This distribution $P(t_{\rm m}|T)$ becomes uniform in the "bulk" $1\ll t_{\rm m}\ll T$ and has a nontrivial universal shape in the "edge regimes" $t_{\rm m}\to0$ and $t_{\rm m} \to T$. Some of these results have been announced in a recent Letter [Europhys. Lett. {\bf 135}, 30003 (2021)]., Comment: 57 pages, 19 figures. This is a longer version of arXiv:2104.07346, published in Europhysics Letters

We investigate the number $V_p(n)$ of distinct sites visited by an $n$-step resetting random walker on a $d$-dimensional hypercubic lattice with resetting probability $p$. In the case $p=0$, we recover the well-known result that the average number of distinct sites grows for large $n$ as $\langle V_0(n)\rangle\sim n^{d/2}$ for $d<2$ and as $\langle V_0(n)\rangle\sim n$ for $d>2$. For $p>0$, we show that $\langle V_p(n)\rangle$ grows extremely slowly as $\sim \left[\log(n)\right]^d$. We observe that the recurrence-transience transition at $d=2$ for standard random walks (without resetting) disappears in the presence of resetting. In the limit $p\to 0$, we compute the exact crossover scaling function between the two regimes. In the one-dimensional case, we derive analytically the full distribution of $V_p(n)$ in the limit of large $n$. Moreover, for a one-dimensional random walker, we introduce a new observable, which we call imbalance, that measures how much the visited region is symmetric around the starting position. We analytically compute the full distribution of the imbalance both for $p=0$ and for $p>0$. Our theoretical results are verified by extensive numerical simulations., Comment: 41 pages, 8 figs

``When in a difficult situation, it is sometimes better to give up and start all over again''. While this empirical truth has been regularly observed in a wide range of circumstances, quantifying the effectiveness of such a heuristic strategy remains an open challenge. In this paper, we combine the notions of optimal control and stochastic resetting to address this problem. The emerging analytical framework allows not only to measure the performance of a given restarting policy but also to obtain the optimal strategy for a wide class of dynamical systems. We apply our technique to a system with a final reward and show that the reward value must be larger than a critical threshold for resetting to be effective. Our approach, analogous to the celebrated Hamilton-Jacobi-Bellman paradigm, provides the basis for the investigation of realistic restarting strategies across disciplines. As an application, we show that the framework can be applied to an epidemic model to predict the optimal lockdown policy., Comment: 11 pages, 7 figures

We consider a single run-and-tumble particle (RTP) moving in one dimension. We assume that the velocity of the particle is drawn independently at each tumbling from a zero-mean Gaussian distribution and that the run times are exponentially distributed. We investigate the probability distribution $P(X,N)$ of the position $X$ of the particle after $N$ runs, with $N\gg 1$. We show that in the regime $ X \sim N^{3/4}$ the distribution $P(X,N)$ has a large deviation form with a rate function characterized by a discontinuous derivative at the critical value $X=X_c>0$. The same is true for $X=-X_c$ due to the symmetry of $P(X,N)$. We show that this singularity corresponds to a first-order condensation transition: for $X>X_c$ a single large jump dominates the RTP trajectory. We consider the participation ratio of the single-run displacements as the order parameter of the system, showing that this quantity is discontinuous at $X=X_c$. Our results are supported by numerical simulations performed with a constrained Markov chain Monte Carlo algorithm., Comment: 24 pages, 10 figures

We consider a one-dimensional stationary stochastic process $x(\tau)$ of duration $T$. We study the probability density function (PDF) $P(t_{\rm m}|T)$ of the time $t_{\rm m}$ at which $x(\tau)$ reaches its global maximum. By using a path integral method, we compute $P(t_{\rm m}|T)$ for a number of equilibrium and nonequilibrium stationary processes, including the Ornstein-Uhlenbeck process, Brownian motion with stochastic resetting and a single confined run-and-tumble particle. For a large class of equilibrium stationary processes that correspond to diffusion in a confining potential, we show that the scaled distribution $P(t_{\rm m}|T)$, for large $T$, has a universal form (independent of the details of the potential). This universal distribution is uniform in the ``bulk'', i.e., for $0 \ll t_{\rm m} \ll T$ and has a nontrivial edge scaling behavior for $t_{\rm m} \to 0$ (and when $t_{\rm m} \to T$), that we compute exactly. Moreover, we show that for any equilibrium process the PDF $P(t_{\rm m}|T)$ is symmetric around $t_{\rm m}=T/2$, i.e., $P(t_{\rm m}|T)=P(T-t_{\rm m}|T)$. This symmetry provides a simple method to decide whether a given stationary time series $x(\tau)$ is at equilibrium or not., Comment: Main text: 6 pages + 4 figs., Supp. Mat.: 7 pages + 3 figs

We study the position distribution $P(\vec{R},N)$ of a run-and-tumble particle (RTP) in arbitrary dimension $d$, after $N$ runs. We assume that the constant speed $v>0$ of the particle during each running phase is independently drawn from a probability distribution $W(v)$ and that the direction of the particle is chosen isotropically after each tumbling. The position distribution is clearly isotropic, $P(\vec{R},N)\to P(R,N)$ where $R=|\vec{R}|$. We show that, under certain conditions on $d$ and $W(v)$ and for large $N$, a condensation transition occurs at some critical value of $R=R_c\sim O(N)$ located in the large deviation regime of $P(R,N)$. For $RR_c$ is typically dominated by a `condensate', i.e., a large single run that subsumes a finite fraction of the total displacement (supercritical condensed phase). Focusing on the family of speed distributions $W(v)=\alpha(1-v/v_0)^{\alpha-1}/v_0$, parametrized by $\alpha>0$, we show that, for large $N$, $P(R,N)\sim \exp\left[-N\psi_{d,\alpha}(R/N)\right]$ and we compute exactly the rate function $\psi_{d,\alpha}(z)$ for any $d$ and $\alpha$. We show that the transition manifests itself as a singularity of this rate function at $R=R_c$ and that its order depends continuously on $d$ and $\alpha$. We also compute the distribution of the condensate size for $R>R_c$. Finally, we study the model when the total duration $T$ of the RTP, instead of the total number of runs, is fixed. Our analytical predictions are confirmed by numerical simulations, performed using a constrained Markov chain Monte Carlo technique, with precision $\sim 10^{-100}$., Comment: 44 pages, 14 figures

We compute exactly the mean perimeter and the mean area of the convex hull of a $2$-d Brownian motion of duration $t$ and diffusion constant $D$, in the presence of resetting to the origin at a constant rate $r$. We show that for any $t$, the mean perimeter is given by $\langle L(t)\rangle= 2 \pi \sqrt{\frac{D}{r}}\, f_1(rt)$ and the mean area is given by $\langle A(t) \rangle= 2\pi\frac{D}{r}\, f_2(rt)$ where the scaling functions $f_1(z)$ and $f_2(z)$ are computed explicitly. For large $t\gg 1/r$, the mean perimeter grows extremely slowly as $\langle L(t)\rangle \propto \ln (rt)$ with time. Likewise, the mean area also grows slowly as $\langle A(t)\rangle \propto \ln^2(rt)$ for $t\gg 1/r$. Our exact results indicate that the convex hull, in the presence of resetting, approaches a circular shape at late times. Numerical simulations are in perfect agreement with our analytical predictions., Comment: 16 pages, 6 figures. Comments added about the circular shape of the convex hull at large time

We consider a model of space-continuous one-dimensional random walk with simple correlation between the steps: the probability that two consecutive steps have same sign is $q$ with $0\leq q\leq 1$. The parameter $q$ allows thus to control the persistence of the random walk. We compute analytically the survival probability of a walk of $n$ steps, showing that it is independent of the jump distribution for any finite $n$. This universality is a consequence of the Sparre-Andersen theorem for random walks with uncorrelated and symmetric steps. We then apply this result to derive the distribution of the step at which the random walk reaches its maximum and the record statistics of the walk, which show the same universality. In particular, we show that the distribution of the number of records for a walk of $n\gg 1$ steps is the same as for a random walk with $n_{\rm eff}(q)=n/(2(1-q))$ uncorrelated and symmetrically distributed steps. We also show that in the regime where $n\to \infty$ and $q\to 1$ with $y=n(1-q)$, this model converges to the run-and-tumble particle, a persistent random walk often used to model the motion of bacteria. Our theoretical results are confirmed by numerical simulations., Comment: 28 pages, 10 figures

We consider an active run-and-tumble particle (RTP) in $d$ dimensions, starting from the origin and evolving over a time interval $[0,t]$. We examine three different models for the dynamics of the RTP: the standard RTP model with instantaneous tumblings, a variant with instantaneous runs and a general model in which both the tumblings and the runs are non-instantaneous. For each of these models, we use the Sparre Andersen theorem for discrete-time random walks to compute exactly the probability that the $x$ component does not change sign up to time $t$, showing that it does not depend on $d$. As a consequence of this result, we compute exactly other $x$-component properties, namely the distribution of the time of the maximum and the record statistics, showing that they are universal, i.e. they do not depend on $d$. Moreover, we show that these universal results hold also if the speed $v$ of the particle after each tumbling is random, drawn from a generic probability distribution. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 124, 090603 (2020)]., Comment: 34 pages, 18 figures. This is an extended version of arXiv:2001.01492, published in Physical Review Letters

We consider a one-dimensional Brownian motion of fixed duration $T$. Using a path-integral technique, we compute exactly the probability distribution of the difference $\tau=t_{\min}-t_{\max}$ between the time $t_{\min}$ of the global minimum and the time $t_{\max}$ of the global maximum. We extend this result to a Brownian bridge, i.e. a periodic Brownian motion of period $T$. In both cases, we compute analytically the first few moments of $\tau$, as well as the covariance of $t_{\max}$ and $t_{\min}$, showing that these times are anti-correlated. We demonstrate that the distribution of $\tau$ for Brownian motion is valid for discrete-time random walks with $n$ steps and with a finite jump variance, in the limit $n\to \infty$. In the case of L\'evy flights, which have a divergent jump variance, we numerically verify that the distribution of $\tau$ differs from the Brownian case. For random walks with continuous and symmetric jumps we numerically verify that the probability of the event "$\tau = n$" is exactly $1/(2n)$ for any finite $n$, independently of the jump distribution. Our results can be also applied to describe the distance between the maximal and minimal height of $(1+1)$-dimensional stationary-state Kardar-Parisi-Zhang interfaces growing over a substrate of finite size $L$. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 123, 200201 (2019)]., Comment: 33 pages, 12 figures. This is a longer version of arXiv:1909.05594, published in Physical Review Letters. Typos corrected

We consider an active run-and-tumble particle (RTP) in $d$ dimensions and compute exactly the probability $S(t)$ that the $x$-component of the position of the RTP does not change sign up to time $t$. When the tumblings occur at a constant rate, we show that $S(t)$ is independent of $d$ for any finite time $t$ (and not just for large $t$), as a consequence of the celebrated Sparre Andersen theorem for discrete-time random walks in one dimension. Moreover, we show that this universal result holds for a much wider class of RTP models in which the speed $v$ of the particle after each tumbling is random, drawn from an arbitrary probability distribution. We further demonstrate, as a consequence, the universality of the record statistics in the RTP problem., Comment: Main text: 5 pages + 2 figs., Supp. Mat: 12 pages + 4 figs

We present an exact solution for the probability density function $P(\tau=t_{\min}-t_{\max}|T)$ of the time-difference between the minimum and the maximum of a one-dimensional Brownian motion of duration $T$. We then generalise our results to a Brownian bridge, i.e. a periodic Brownian motion of period $T$. We demonstrate that these results can be directly applied to study the position-difference between the minimal and the maximal height of a fluctuating $(1+1)$-dimensional Kardar-Parisi-Zhang interface on a substrate of size $L$, in its stationary state. We show that the Brownian motion result is universal and, asymptotically, holds for any discrete-time random walk with a finite jump variance. We also compute this distribution numerically for L\'evy flights and find that it differs from the Brownian motion result., Comment: Main text (published version): 5 pages + 3 Figs, Supp. Mat.: 20 pages + 7 Figs, typos corrected