[en] We consider a system of two Brownian particles (say A and B), coupled to each other via harmonic potential of stiffness constant k. Particle-A is connected to two heat baths of constant temperatures T ₁ and T ₂, and particle-B is connected to a single heat bath of a constant temperature T ₃. In the steady state, the total entropy production for both particles obeys the fluctuation theorem. We compute the total entropy production due to particle-A in the coupled system (partial and apparent entropy production) in the steady state for a time segment τ. When both particles are weakly interacting with each other, the fluctuation theorem for partial and apparent entropy production is studied. We find a significant deviation from the fluctuation theorem. The analytical results are also verified using numerical simulations. Furthermore, we investigate the effect of hidden fast degrees of freedom on the steady state fluctuation theorem for a system of a single Brownian particle coupled to two heat baths of distinct temperatures and dissipation constants, and coupled to a third heat bath of vanishing temperature and dissipation constant. (paper: classical statistical mechanics, equilibrium and non-equilibrium)

[en] We study the effect of stochastic resetting on a run-and-tumble particle (RTP) in two spatial dimensions. We consider a resetting protocol which affects both the position and orientation of the RTP: the particle undergoes constant-rate positional resetting to a fixed point in space and a random orientation. We compute the radial and x-marginal stationary-state distributions and show that while the former approaches a constant value as r → 0, the latter diverges logarithmically as x → 0. On the other hand, both the marginal distributions decay exponentially with the same exponent when they are far from the origin. We also study the temporal relaxation of the RTP and show that the positional distribution undergoes a dynamic transition to a stationary state. We also study the first-passage properties of the RTP in the presence of resetting and show that the optimization of the resetting rate can minimize the mean first-passage time. We also provide a brief discussion of the stationary states for resetting a particle to an initial position with a fixed orientation. (paper)

[en] We investigate the run-and-tumble particle (RTP), also known as persistent Brownian motion, in one dimension. A telegraphic noise σ(t) drives the particle which changes between ±1 values at certain rates. Denoting the rate of flip from 1 to −1 as R ₁ and the converse rate as R ₂, we consider the position- and direction-dependent rates of the form $R_1\left(x\right)={\left(\frac{|x|}{l}\right)}^{\mathrm{\alpha <!--\; ??\; -->}}[{\mathrm{\gamma <!--\; ??\; -->}}_1\mathrm{\theta <!--\; ??\; -->}\left(x\right)+{\mathrm{\gamma <!--\; ??\; -->}}_2\mathrm{\theta <!--\; ??\; -->}(-<!--\; ???\; -->x)]$ and $R_2\left(x\right)={\left(\frac{|x|}{l}\right)}^{\mathrm{\alpha <!--\; ??\; -->}}[{\mathrm{\gamma <!--\; ??\; -->}}_2\mathrm{\theta <!--\; ??\; -->}\left(x\right)+{\mathrm{\gamma <!--\; ??\; -->}}_1\mathrm{\theta <!--\; ??\; -->}(-<!--\; ???\; -->x)]$ with α ⩾ 0. For α = 0 and 1, we solve the master equations exactly for arbitrary γ ₁ and γ ₂ at large t. From our analytical expression for the time-dependent probability distribution P(x, t) we find that for γ ₁ > γ ₂ the distribution relaxes to a steady state exponentially, whereas for γ ₁ ⩽ γ ₂ the distribution does not reach a steady state and can be described by a non-trivial scaling form. We interestingly find that these features of the probability distribution P(x, t) in the two regimes γ ₁ > γ ₂ and γ ₁ ⩽ γ ₂ also remain valid for general α > 0. In particular, for general α, we argue and numerically demonstrate that the approach to the steady state in γ ₁ > γ ₂ case is exponential. On the other hand, for γ ₁ ⩽ γ ₂, the distribution P(x, t) remains time dependent and possesses certain scaling behavior. For γ ₁ = γ ₂ we derive the scaling behavior as well as the scaling function rigorously, whereas for γ ₁ < γ ₂ we provide heuristic arguments to obtain the scaling behavior and the corresponding scaling functions. We also study the dynamics on a semi-infinite line with an absorbing barrier at the origin. For α = 0 and 1, we analytically compute the survival probabilities and the corresponding first-passage time distributions. For general α ⩾ 0, we provide approximate calculations to compute the behavior of the survival probability for t → ∞ in which limit it approaches a finite value for γ ₁ < γ ₂ but goes to zero for γ ⩾ γ ₂. We also study the approach to the large t value in both cases. Finally, we consider RTP in a finite interval [0, M] and compute the associated exit probabilities from that interval for all α. All our analytic results are verified with a numerical simulation. (paper)

[en] We consider a particle in a one-dimensional box of length L, with a Maxwell bath at one end and a reflecting wall at the other end. Using a renewal approach, as well as directly solving the master equation, we show that the system exhibits a slow power law relaxation, with a logarithmic correction, towards the final equilibrium state. We extend the renewal approach to a class of confining potentials of the form $U(x)\propto <!--\; ???\; -->x^{\mathrm{\alpha <!--\; ??\; -->}}$, $x>0$, where we find that the relaxation is $\sim <!--\; ???\; -->t^{-<!--\; ???\; -->(\mathrm{\alpha <!--\; ??\; -->}+2)/(\mathrm{\alpha <!--\; ??\; -->}-<!--\; ???\; -->2)}$ for $\mathrm{\alpha <!--\; ??\; -->}>2$, with a logarithmic correction when $(\mathrm{\alpha <!--\; ??\; -->}+2)/(\mathrm{\alpha <!--\; ??\; -->}-<!--\; ???\; -->2)$ is an integer. For $\mathrm{\alpha <!--\; ??\; -->}<2$ the relaxation is exponential. Interestingly for $\mathrm{\alpha <!--\; ??\; -->}=2$ (harmonic potential) the localised bath cannot equilibrate the particle. (paper: classical statistical mechanics, equilibrium and non-equilibrium)

[en] We consider a gas of point particles moving on the one-dimensional line with a hard-core inter-particle interaction that prevents particle crossings—this is usually referred to as single-file motion. The individual particle dynamics can be arbitrary and they only interact when they meet. Starting from initial conditions such that particles are uniformly distributed, we observe the displacement of a tagged particle at time t, with respect to the initial position of another tagged particle, such that their tags differ by r. For r   =   0, this is the usual well studied problem of the tagged particle motion. Using a mapping to a non-interacting particle system we compute the exact probability distribution function for the two-tagged particle displacement, for general single particle dynamics. As by-products, we compute the large deviation function, various cumulants and, for the case of Hamiltonian dynamics, the two-particle velocity auto-correlation function

[en] The probability distribution of the total entropy production in the non-equilibrium steady state follows a symmetry relation called the fluctuation theorem. When a certain part of the system is masked or hidden, it is difficult to infer the exact estimate of the total entropy production. Entropy produced from the observed part of the system shows significant deviation from the steady state fluctuation theorem. This deviation occurs due to the interaction between the observed and the masked part of the system. A naive guess would be that the deviation from the steady state fluctuation theorem may disappear in the limit of small interaction between both parts of the system. In contrast, we investigate the entropy production of a particle in a harmonically coupled Brownian particle system (say, particle A and B) in a heat reservoir at a constant temperature. The system is maintained in the non-equilibrium steady state using stochastic driving. When the coupling between particle A and B is infinitesimally weak, the deviation from the steady state fluctuation theorem for the entropy production of a partial system of a coupled system is studied. Furthermore, we consider a harmonically confined system (i.e. a harmonically coupled system of particle A and B in harmonic confinement). In the weak coupling limit, the entropy produced by the partial system (e.g. particle A) of the coupled system in a harmonic trap satisfies the steady state fluctuation theorem. Numerical simulations are performed to support the analytical results. Part of these results were announced in a recent letter, see Gupta and Sabhapandit (2016 Europhys. Lett. 115 60003). (paper: classical statistical mechanics, equilibrium and non-equilibrium)

[en] We present exact spatio-temporal correlation functions of a Rouse polymer chain submerged in a fluid having planar mixed flow, in the steady state. Using these correlators, determination of the time scale distribution functions associated with the first-passage tumbling events is difficult in general; it was done recently in Phys. Rev. Lett. 101, 188301 (2008), for the special case of 'simple shear' flow. We show here that the method used in latter paper fails for the general mixed flow problem. We also give many new estimates of the exponent θ associated with the exponential tail of the angular tumbling time distribution in the case of simple shear.

[en] We study the motion of a one-dimensional run-and-tumble particle with three discrete internal states in the presence of a harmonic trap of stiffness The three internal states, corresponding to positive, negative and zero velocities respectively, evolve following a jump process with rate . We compute the stationary position distribution exactly for arbitrary values of and which turns out to have a finite support on the real line. We show that the distribution undergoes a shape-transition as is changed. For the distribution has a double-concave shape and shows algebraic divergences with an exponent both at the origin and at the boundaries. For the position distribution becomes convex, vanishing at the boundaries and with a single, finite, peak at the origin. We also show that for the special case the distribution shows a logarithmic divergence near the origin while saturating to a constant value at the boundaries. (letter)

[en] We consider heat transport across a harmonic chain connected at its two ends to white-noise Langevin reservoirs at different temperatures. In the steady state of this system the heat Q flowing from one reservoir into the system in a finite time τ has a distribution P(Q, τ). We study the large time form of the corresponding moment generating function (e^−λQ) ∼ g(λ)e^τμ(λ). Exact formal expressions, in terms of phonon Green's functions, are obtained for both μ(λ) and also the lowest order correction g(λ). We point out that, in general, a knowledge of both μ(λ) and g(λ) is required for finding the large deviation function associated with P(Q, τ). The function μ(λ) is known to be the largest eigenvector of an appropriate Fokker–Planck type operator and our method also gives the corresponding eigenvector exactly

[en] We study a gas of point particles with hard-core repulsion in one dimension where the particles move freely in-between elastic collisions. We prepare the system with a uniform density on the infinite line. The velocities of the particles are chosen independently from a thermal distribution. Using a mapping to the non-interacting gas, we analytically compute the equilibrium spatio-temporal correlations for arbitrary integers . The analytical results are verified with microscopic simulations of the Hamiltonian dynamics. The correlation functions have ballistic scaling, as expected in an integrable model. (paper: classical statistical mechanics, equilibrium and non-equilibrium)