We establish a refined version of the Second Law of Thermodynamics for Langevin stochastic processes describing mesoscopic systems driven by conservative or non-conservative forces and interacting with thermal noise. The refinement is based on the Monge-Kantorovich optimal mass transport. General discussion is illustrated by numerical analysis of a model for micron-size particle manipulated by optical tweezers.

Thermodynamics of small systems has become an important field of statistical physics. Such systems are driven out of equilibrium by a control, and the question is naturally posed how such a control can be optimized. We show that optimization problems in small system thermodynamics are solved by (deterministic) optimal transport, for which very efficient numerical methods have been developed, and of which there are applications in cosmology, fluid mechanics, logistics, and many other fields. We show, in particular, that minimizing expected heat released or work done during a nonequilibrium transition in finite time is solved by the Burgers equation and mass transport by the Burgers velocity field. Our contribution hence considerably extends the range of solvable optimization problems in small system thermodynamics.

In this paper we study some aspects of search for an immobile target by a swarm of N non-communicating, randomly moving searchers (numbered by the index k, k = 1, 2, . . . , N ), which all start their random motion simultaneously at the same point in space. For each realization of the search process, we record the unordered set of time moments {τ k }, where τ k is the time of the first passage of the k-th searcher to the location of the target. Clearly, τ k 's are independent, identically distributed random variables with the same distribution function Ψ(τ ). We evaluate then the distribution P (ω) of the random variable ω ∼ τ 1 /τ , where τ = N −1 N k=1 τ k is the ensembleaveraged realization-dependent first passage time. We show that P (ω) exhibits quite a non-trivial and sometimes a counterintuitive behaviour. We demonstrate that in some well-studied cases (e.g., Brownian motion in finite d-dimensional domains) the mean first passage time is not a robust measure of the search efficiency, despite the fact that Ψ(τ ) has moments of arbitrary order. This implies, in particular, that even in this simplest case (not saying about complex systems and/or anomalous diffusion) first passage data extracted from a single particle tracking should be regarded with an appropriate caution because of the significant sample-to-sample fluctuations. PACS numbers: 02.50.-r, 05.40.-a, 87.10.Mn

We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability P (ω) distribution of the random variable ω = τ1/(τ1 +τ2), which is a measure for how similar the first passage times τ1 and τ2 are of two independent realisations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether P (ω) represents a unimodal, bell-shaped form, or a bimodal, M-shaped behavior. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behavior, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of P (ω), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in twodimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behavior even in bounded domains, in which all moments of the first passage distribution exist.