We explore the thermodynamic geometry of a simple system that models the bistable dynamics of nucleic acid hairpins in single molecule force-extension experiments. Near equilibrium, optimal (minimum-dissipation) driving protocols are governed by a generalized linear response friction coefficient. Our analysis and simulations demonstrate that the friction coefficient of the driving protocols is sharply peaked at the interface between metastable regions, which leads to minimum-dissipation protocols that drive rapidly within a metastable basin, but then linger longest at the interface, giving thermal fluctuations maximal time to kick the system over the barrier. Intuitively, the same principle applies generically in free energy estimation (both in steered molecular dynamics simulations and in single-molecule experiments), provides a design principle for the construction of thermodynamically efficient coupling between stochastic objects, and makes a prediction regarding the construction of evolved biomolecular motors.
In this paper we have presented the mechanism of the barrier crossing dynamics of a Brownian particle which is coupled to a thermal bath in the presence of both time independent and fluctuating magnetic fields. Here the following three aspects are important in addition to the role of the thermal bath on the barrier crossing dynamics. Magnetic field induced coupling may introduce a resonance like effect. Another role of the field is that enhancement of its strength reduces the frequency factor of the barrier crossing rate constant. Finally, the fluctuating magnetic field introduces an induced e…
By using the backward fractional Fokker-Planck equation we investigate the barrier crossing event in the presence of Levy noise. After shortly review recent results obtained with different approaches on the time characteristics of the barrier crossing, we derive a general differential equation useful to calculate the nonlinear relaxation time. We obtain analytically the nonlinear relaxation time for free Levy flights and a closed expression in quadrature of the same characteristics for cubic potential.
Driven barrier crossings are pervasive in optical-trapping experiments and steered molecular-dynamics simulations. Despite the high fidelity of control, the freedom in the choice of driving protocol is rarely exploited to improve efficiency. We design protocols that reduce dissipation for rapidly driven barrier crossing under two-dimensional control of a harmonic trapping potential, controlling both trap center and stiffness. For fast driving, the minimum-dissipation protocol jumps halfway between the control-parameter endpoints. For slow driving, the minimum-dissipation protocol generically s…
We study a system of non-interacting active particles, propelled by colored noises, characterized by an activity time $τ$, and confined by a double-well potential. A straightforward application of this system is the problem of barrier crossing of active particles, which has been studied only in the limit of small activity. When $τ$ is sufficiently large, equilibrium-like approximations break down in the barrier crossing region. In the model under investigation, it emerges a sort of "negative temperature" region, and numerical simulations confirm the presence of non-convex local velocity distri…
We explore the dependence of the thermally activated barrier crossing rate on various model parameters for a dimer that undergoes a Brownian motion on a piecewise linear bistable potential employing the method of adiabatic elimination of fast variable. By introducing a different model system and approaches than the previous works \cite{c4,c5}, not only we recapture the previous results but we further show that systematic elimination of the fast changing variable leads to an effective Kramers type potential. It is shown that for rigid dimer, the escape rate $R$ monotonously decreases with $k$. …
We study the stochastic resonance (SR) of a flexible polymer chain crossing over a piecewise linear bistable potential. The dependence of signal to noise ratio $SNR$ on noise intensity $D$, coupling constant $k$ and polymer length $N$ is studied via two state approximation. We find that the response of signal to the background noise strength is significant at optimum values of $D_{opt}$, $k_{opt}$ and $N_{opt}$ which suggests novel means of manipulating proteins or vesicles. Furthermore, the thermally activated barrier crossing rate $r_{k}$ for the flexible polymer chain is studied. We find th…
Barrier crossing calculations in chemical reaction-rate theory typically assume that the barrier is large compared to the temperature. When the barrier vanishes, however, there is a qualitative change in behavior. Instead of crossing a barrier, particles slide down a sloping potential. We formulate a renormalization group description of this noisy saddle-node transition. We derive the universal scaling behavior and corrections to scaling for the mean escape time in overdamped systems with arbitrary barrier height. We also develop an accurate approximation to the full distribution of barrier es…
The generalised Langevin equation with a retarded friction and a double-well potential is solved. The random force is modelled by a multiplicative noise with long jumps. Probability density distributions converge with time to a distribution similar to a Gaussian but tails have a power-law form. Dependence of the mean first passage time on model parameters is discussed. Properties of the stochastic resonance, emerging as a peak in the plot of the spectral amplification against the temperature, are discussed for various sets of the model parameters. The amplification rises with the memory and is…
We study the barrier crossing of a particle driven by white symmetric Levy noise of index $α$ and intensity $DD for three different generic types of potentials: (a) a bistable potential; (b) a metastable potential; and (c) a truncated harmonic potential. For the low noise intensity regime we recover the previously proposed algebraic dependence on $D$ of the characteristic escape time, $T_{\mathrm{esc}}\simeq C(α)/D^{μ(α)}$, where $C(α)$ is a coefficient. It is shown that the exponent $μ(α)$ remains approximately constant, $μ\approx 1$ for $0<α<2$; at $α=2$ the power-law form of $T_{\mathrm{esc…
We study the stochastic resonance (SR) of a flexible polymer surmounting a bistable-potential barrier. Due to the flexibility that can enhance crossing rate and change chain conformations at the barrier, the SR behaviors manifest many features of an entropic SR of a new kind, such as the power amplification peaks at optimal chain lengths and elastic constants as well as the optimal noise strengths. The pronounced peaks that emerge depending on the chain lengths and conformation states suggest novel means of manipulating biopolymers, such as efficient separation methods, within undulating chann…
We study the decay rate $θ(a)$ that characterizes the late time exponential decay of the first-passage probability density $F_a(t|0) \sim e^{-θ(a)\, t}$ of a diffusing particle in a one dimensional confining potential $U(x)$, starting from the origin, to a position located at $a>0$. For general confining potential $U(x)$ we show that $θ(a)$, a measure of the barrier (located at $a$) crossing rate, has three distinct behaviors as a function of $a$, depending on the tail of $U(x)$ as $x\to -\infty$. In particular, for potentials behaving as $U(x)\sim |x|$ when $x\to -\infty$, we show that a nove…
A spatially extended classical system with metastable states subject to weak spatiotemporal noise can exhibit a transition in its activation behavior when one or more external parameters are varied. Depending on the potential, the transition can be first or second-order, but there exists no systematic theory of the relation between the order of the transition and the shape of the potential barrier. In this paper, we address that question in detail for a general class of systems whose order parameter is describable by a classical field that can vary both in space and time, and whose zero-noise …
We introduce a generalized Langevin model system for different non-Markovian effects in the well and barrier regions of a potential, and use it to numerically study the dependence of the barrier-crossing time. In the appropriate limits, our model interpolates between the theoretical barrier-crossing-time predictions by Grote and Hynes (GH), as well as by Pollak et al., which for a single barrier memory time can differ by several orders of magnitude. Our model furthermore allows to test an analytic rate theory for space-inhomogeneous memory, which disagrees with our numerical results in the lon…
We numerically study the escape kinetics of a self-propelled Janus particle, carrying a cargo, from a meta-stable state. We assume that the cargo is attached to the Janus particle by a flexible harmonic spring. We take into account the effect of velocity field, created in the fluid due to movements of dimer's components, by considering space-dependent diffusion tensor (Oseen tensor). Our simulation results show that the synchronization between barrier crossing events and rotational relaxation process can enhance escape rate to a large extent. Also, the load carrying capability of a Janus parti…
We consider the motion of semiflexible polymers in double-well potentials. We calculate shape, energy, and effective diffusion constant of kink excitations, and in particular their dependence on the bending rigidity of the semiflexible polymer. For symmetric potentials, the kink motion is purely diffusive whereas kink motion becomes directed in the presence of a driving force on the polymer. We determine the average velocity of the semiflexible polymer based on the kink dynamics. The Kramers escape over the potential barriers proceeds by nucleation and diffusive motion of kink-antikink pairs, …
The problem of computing the rate of diffusion-aided activated barrier crossings between metastable states is one of broad relevance in physical sciences. The transition path formalism aims to compute the rate of these events by analysing the statistical properties of the transition path between the two metastable regions concerned. In this paper, we show that the transition path process is a unique solution to an associated stochastic differential equation (SDE), with a discontinuous and singular drift term. The singularity arises from a local time contribution, which accounts for the fluctua…
The two-dimensional cage model for polymer motion is discussed with an emphasis on the effect of sideways motions, which cross the barriers imposed by the lattice. Using the Density Matrix Method as a solver of the Master Equation, the renewal time and the diffusion coefficient are calculated as a function of the strength of the barrier crossings. A strong crossover influence of the barrier crossings is found and it is analyzed in terms of effective exponents for a given chain length. The crossover scaling functions and the crossover scaling exponents are calculated.
We investigate memory effects in barrier-crossing in the overdamped setting. We focus on the scenario where the hidden degrees of freedom relax on exactly the same time scale as the observable. As a prototypical model, we analyze tagged-particle diffusion in a single file confined to a bi-stable potential. We identify the signatures of memory and explain their origin. The emerging memory is a result of the projection of collective many-body eigenmodes onto the motion of a tagged-particle. We are interested in the "confining" (all background particles in front of the tagged-particle) and "pushi…
Target search by active agents in rugged energy landscapes has remained a challenge because standard enhanced sampling methods do not apply to irreversible dynamics. We overcome this non-equilibrium rare-event problem by developing an algorithm generalizing transition-path sampling to active Brownian dynamics. This method is exemplified and benchmarked for a paradigmatic two-dimensional potential with a high barrier. We find that even in such a simple landscape the structure and kinetics of the ensemble of transition paths change drastically in the presence of activity. Indeed, active Brownian…
We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum mechanical mean values of their co-ordinates and momenta we have derived a $c$-number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equ…
Based on a true phase space probability distribution function and an ensemble averaging procedure we have recently developed [Phys. Rev. E 65, 021109 (2002)] a non-Markovian quantum Kramers' equation to derive the quantum rate coefficient for barrier crossing due to thermal activation and tunneling in the intermediate to strong friction regime. We complement and extend this approach to weak friction regime to derive quantum Kramers' equation in energy space and the rate of decay from a metastable well. The theory is valid for arbitrary temperature and noise correlation. We show that depending …
Biomolecular conformational transitions are usually modeled as barrier crossings in a free energy landscape. The transition paths connect two local free energy minima and transition path times (TPT) are the actual durations of the crossing events. The simplest model employed to analyze TPT and to fit empirical data is that of a stochastic particle crossing a parabolic barrier. Motivated by some disagreement between the value of the barrier height obtained from the TPT distributions as compared to the value obtained from kinetic and thermodynamic analyses, we investigate here TPT for barriers w…
We study the influence of thermal fluctuations on the buckling behavior of thin elastic capsules with spherical rest shape. Thermal fluctuations affect the buckling instability by two mechanisms. On the one hand, thermal fluctuations can renormalize the capsule's elastic properties and its pressure because of anharmonic couplings between normal displacement modes of different wavelengths. This effectively lowers its critical buckling pressure [Kosmrlj and Nelson, Phys. Rev. X 7, 011002 (2017)]. On the other hand, buckled shapes are energetically favorable already at pressures below the classic…
We consider the generalization of the Kramers escape over a barrier problem to the case of a long chain molecule. The problem involves the motion of a chain molecule of $N$ segments across a region where the free energy per segment is higher, so that it has to cross a barrier. We consider the limit where the length of the molecule is much larger than the width of the barrier. The width is taken to be sufficiently wide that a coninuum description is applicable to even to the portion over the barrier. We use the Rouse model and analyze the mechanism of crossing a barrier. There can be two domina…
We introduce a switching mechanism in the asymptotic occupations of quantum states induced by the combined effects of a periodic driving and a weak coupling to a heat bath. It exploits one of the ubiquitous avoided crossings in driven systems and works even if both involved Floquet states have small occupations. It is independent of the initial state and the duration of the driving. As a specific example of this general switching mechanism we show how an asymmetric double well potential can be switched between the lower and the upper well by a periodic driving that is much weaker than the asym…
We consider the motion of a particle in a force field subjected to adiabatic, fluctuations of external origin. We do not put the restriction on the type of stochastic process that the noise is Gaussian. Based on a method developed earlier by us [ J. Phys. A {\bf 31} (1998) 3937, 7301] we have derived the equation of motion for probability distribution function for the particle on a coarse-grained timescale $Δt$ assuming that it satisfies the separation of timescales; $|μ|^{-1} \ll Δt \ll τ_c$, where $τ_c$ is the correlation time of fluctuations. $|μ|^{-1}$ refers to the inverse of the damping …
We establish a stochastic thermodynamics for a Fermionic level driven by a time-dependent force and interacting with initially thermalized levels playing the role of a reservoir. The driving induces consecutive avoided crossings between system and reservoir levels described within Landau-Zener theory. We derive the resulting system dynamics and thermodynamics and identify energy, work, heat, entropy and dissipation. Our theory perfectly reproduces the numerically exact quantum work statistics obtained using a two point measurements approach of the total energy and provides an explicit expressi…
An alternative scenario for the glass transition based on the cooperative nature of nucleation processes and the role of entropic effects is presented. The new ingredient is to relate the dissipation during the relaxation process to the release of strain energy driven by the nucleation of progressively larger cooperative spatial regions. Using an equiprobability hypothesis for the transition between different metastable configurations, we obtain a relation between the free energy dissipation rate and the size of the largest cooperative regions. Th is leads to a new phenomenological relation be…
We compute analytically the probability $S(t)$ that a set of $N$ Brownian paths do not cross each other and stay below a moving boundary $g(τ)= W \sqrtτ$ up to time $t$. We show that for large $t$ it decays as a power law $S(t) \sim t^{- β(N,W)}$. The decay exponent $β(N,W)$ is obtained as the ground state energy of a quantum system of $N$ non-interacting fermions in a harmonic well in the presence of an infinite hard wall at position $W$. Explicit expressions for $β(N,W)$ are obtained in various limits of $N$ and $W$, in particular for large $N$ and large $W$. We obtain the joint distribution…
The dielectric response of non-interacting dipoles is discussed in the framework of the classical model of stochastic reorientations in an asymmetric double well potential (ADWP). In the nonlinear regime, this model exhibits some pecularities in the static response. We find that the saturation behavior of the symmetric double well potential model does not follow the Langevin function and only in the linear regime the standard results are recovered. If a finite asymmetry is assumed, the nonlinear susceptibilities are found to change the sign at a number of characteristic temperatures that depen…
We focus on the energy landscape of a simple mean-field model of glasses and analyze activated barrier-crossing by combining the Kac-Rice method for high-dimensional Gaussian landscapes with dynamical field theory. In particular, we consider Langevin dynamics at low temperature in the energy landscape of the pure spherical $p$-spin model. We select as initial condition for the dynamics one of the many unstable index-1 saddles in the vicinity of a reference local minimum. We show that the associated dynamical mean-field equations admit two solutions: one corresponds to falling back to the origi…
The time evolution of the thermally activated decay rates is considered. This evolution is of particular importance for the recent nanoscale experiments discussed in the literature, where the potential barrier is relatively low (or the temperature is relatively high). The single-molecule pulling is one example of such experiments. The decay process is modeled in the present work through computer solving the stochastic (Langevin) equations. Altogether about a hundred of high precision rates have been obtained and analyzed. The rates are registered at the absorption point located far beyond the …
At molecular scales, fluctuations play a significant role and prevent biomolecular processes from always proceeding in a preferred direction, raising the question of how limited amounts of free energy can be dissipated to obtain directed progress. We examine the system and process characteristics that efficiently break time-reversal symmetry at fixed energy loss; in particular for a simple model of a molecular machine, an intermediate energy barrier produces unusually high asymmetry for a given dissipation. We relate the symmetry-breaking factors found in this model to recent observations of b…
The over-passing probability across an inverted parabolic potential barrier is investigated according to the classical and quantal generalized Langevin equations. It is shown that, in the classical case, the asymptotic value of the over-passing probability is determined by a single dominant root of the "characteristic function", and it is given by a simple expression. The expression for the over-passing probability is quite general, and details of dissipation mechanism and memory effects enter into the expression only through the dominant root of the characteristic equation.
Non-equilibrium critical phenomena generally exist in many dynamic systems, like chemical reactions and some driven-dissipative {reactive} particle systems. Here, by using computer simulation and theoretical analysis, we demonstrate the crucial role of the activation barrier on the criticality of dynamic phase transitions in a minimal reactive hard-sphere model. We find that at zero thermal noise, with increasing the activation barrier, the type of transition changes from a continuous conserved directed percolation into a discontinuous dynamic transition by crossing a \emph{tricritical} point.…
Biological macromolecules have complex and non-trivial energy landscapes, endowing them a unique conformational adaptability and diversity in function. Hence, understanding the processes of elasticity and dissipation at the nanoscale is important to molecular biology and also emerging fields such as nanotechnology. Here we analyse single molecule fluctuations in an atomic force microscope (AFM) experiment using a generic model of biopolymer viscoelasticity that importantly includes sources of local `internal' conformational dissipation. Comparing two biopolymers, dextran and cellulose, polysac…
Single-molecule experiments have found near-perfect thermodynamic efficiency in the rotary motor F1-ATP synthase. To help elucidate the principles underlying nonequilibrium energetic efficiency in such stochastic machines, we investigate driving protocols that minimize dissipation near equilibrium in a simple model rotary mechanochemical motor, as determined by a generalized friction coefficient. Our simple model has a periodic friction coefficient that peaks near system energy barriers. This implies a minimum-dissipation protocol that proceeds rapidly when the system is overwhelmingly in a si…
