On the analysis of the integral fluctuation theorem for accelerated colloidal systems in the long-time limit


 A quantitative description of the violation of the second law of thermodynamics in relatively small classical systems and over short time scales comes from the fluctuation-dissipation theorem. It has been well established both theoretically and experimentally, the validity of the fluctuation theorem to small scale systems that are disturbed from their initial equilibrium states. Some experimental studies in the past have also explored the validity of the fluctuation theorem to nonequilibrium steady states at long time scales in the 
asymptotic limit. To this end, a theoretical and/or purely numerical model of the integral fluctuation theorem has been presented. An approximate general expression for the dissipation function has been derived for accelerated colloidal systems trapped/confined in power-law traps. Thereafter, a colloidal particle trapped in a harmonic potential (generated by an accelerating one-dimensional optical trap) and undergoing Brownian motion has been considered for the numerical study. A toy model of a quartic potential trap in addition to the harmonic trap has also been considered for the numerical study. The results presented herein show that the integral fluctuation theorem applies not only to equilibrium steady state distributions but also to 
nonequilibrium steady state distributions of colloidal systems in accelerated frames of reference over long time scales.


Introduction
The behavior of classical systems under the influence of external perturbations and microscopic forces (both, directed and unavoidable) has been a subject of long-standing interest amongst both, physicists and chemists alike. They seek to quantify the dynamics of such systems by accounting for the effects of all external perturbations and ascertain if a closed system of equations of motions can be derived that convincingly describe the behavior of the system, at least up to a reasonable approximation. Newton's equations of motion for instance, are time reversible, by which it is implied that for every classical trajectory, it is possible to construct a time-reversed trajectory that also serves as a solution to the classical equations of motion [1]. In other words, these timereversed trajectories signify the theoretical reversibility of a system. This implies that the higher the probability of observing time-reversed trajectories in a classical system, the greater will be its 'reversibility'. In this specific context, the standard form of the second law of thermodynamics states that a thermodynamic system always evolves in a time-* Department of Physics, Indian Institute of Technology Delhi, New Delhi, India. Email ID: ph1180857@iitd.ac.in.
forward direction in an irreversible manner, i.e., no time-reversed trajectories can be observed in a system. It must be noted however, that the second law of thermodynamics in its most general sense applies solely to large, macroscopic systems over time scales that are much longer as compared to those that we consider in the micro-and the nano-scale length limits. We thus arrive at a very fundamental question that can be stated as: How irreversible are thermodynamic systems of interest at such short length scales? The degree of irreversibility of systems at short length scales can enable them to be considered as promising candidates for the realization of novel nanomachines and biological motors that can essentially be reverse engineered in some sense, i.e., the requirement of external power sources to run such small-scale machines is eliminated, unlike conventional large-scale machines that rely on external power sources for their operation. Very early works of Loschmidt et al. and co-workers focused on deriving irreversible macroscopic equations of motion from reversible equations of motion that are relevant at the microscopic scale. Evans et al. pioneered early works on understanding and quantifying the fluctuation theorem [2] which provides a quantitative description of the temporal evolution of a system's irreversibility from a timereversible system at relatively short time scales to a time-irreversible system (in the thermodynamic limit) at very long time scales. In a more general sense, the fluctuation theorem relates the probabilities of observing conjugate time-reversed trajectories and their corresponding 'time-forward' trajectories of a specific time span, say t which are each characterized by a so-called dissipation function (we denote the dissipation function as t  ). The dissipation function is such that it takes on positive and negative values [1] both, for which the following condition holds for some fixed value  (1) In general, we note that the dissipation function t  assumes a different form for different systems. Eq.
(1) clearly demonstrates that with the increasing passage of time (or an upscaling in the size of the system, important due in part to the fact that t  is extensive), the probability of observing time-reversed trajectories that supposedly violate the second law of thermodynamics becomes increasingly rare and this probability thus dies down to near-zero after a sufficiently long time. Also, given the positivity of the RHS of Eq. (1), we note that the ensemble average of t  is always positive and this holds regardless of the size of the system under consideration. The time scales over which the complete dynamics of the irreversibility of a system can be captured is however not determinable from the fluctuation theorem.
In this paper, we focus our attention to the application of the steady-state fluctuation theorem to a numerical model of a colloidal system that accelerates along one-dimension.  lead to the asymptotic long-time limit in the steady-state fluctuation theorem. An exact form of the dissipation function can be determined as well [1], although in this paper, we solely focus on the derivation and subsequent application of an approximate form of the dissipation function for the colloidal system under consideration. The system that is being considered for the present analysis is such that it would be possible to determine an approximate form for the dissipation function within the realm of deterministic classical dynamics, although an exact, closed-form expression for the dissipation function can be arrived at when one incorporates the framework of Langevin dynamics into the problem at hand (note: we do not concern ourselves with the derivation and analysis of the exact form of the dissipation function in our numerical analysis). The numerical model is based on the drag experiment by Wang et al. [3] in which a colloidal particle was confined/trapped in a harmonic potential generated by a linearly translating optical trap moving at a constant velocity. The numerical analysis presented herein includes an added complexity to the original experiment, i.e., a trap accelerating in the observer's frame of reference has been considered, starting from an initial equilibrium state at time 0 t = . Initially, the colloidal particle is allowed to thermalize in the optical trap, as a consequence of which its position in the trap can be described by an equilibrium Boltzmann distribution, with the average velocity being zero. The trap (and in turn, the colloidal particle) are then disturbed from their initial equilibrium state by setting them in motion along one dimension. After some time, a situation arises wherein the average acceleration of the particle inside the optical trap becomes nearly equal to the trap's acceleration, thus establishing a non-equilibrium steady-state for the system under consideration. The particle's position (on an average) can be determined by balancing the optical force due to the trap and the hydrodynamic force that the colloidal particle experiences due to its movement through the viscous medium. The observer is placed in the frame of the moving colloidal particle, such that the requirement of including effects arising from pseudo-forces can be surmounted. Here, through a numerical analysis, we demonstrate the validity of the fluctuation theorem in the long-time limit (in an accelerating frame of reference) for the case when an approximate form of the dissipation function is considered.
Along the same lines, we then consider the case of a quartic potential trap and demonstrate that the fluctuation theorem holds in the long-time limit when an approximate form of the dissipation function is considered.

A short note on the Langevin equation
For visualizing Brownian motion, we start by implementing a stochastic component to the force acting on the Brownian particle, a description of which is known only probabilistically. For the velocity of the particle, herein denoted as () vt , we have the following first-order differential equation where is the mass of the Brownian particle and the prime denotes the first-order derivative of the velocity v with respect to time. Note that in Eq. A general solution to the Langevin equation is hard to come by, due to the highly non-trivial nature of () ft. In any case however, we are primarily concerned with mean-square values and certain low order correlation functions. We recast Eq. (2) into the form where we define () t  as ( ) We naively state that the stochastic component of the force acting on the Brownian particle is uncorrelated with the velocity v .
However, we must note that a non-zero response of the velocity to () ft does exist, given which we have For 0 tt  , the correlation between () ft and () vt vanishes. We assume now that we have a particle whose velocity is known precisely at time 0 t = . We We know that 0    = . This then gives us the mean of () vt , as For the mean-square velocity of the Brownian particle, we have ..
Note that the in Eq. (7), we eliminate the cross term by making use of the fact that (0 Computing the double integral in Eq. (7) by making an appropriate substitution gives us for the meansquare velocity We note that the mean-square velocity must approach / B k T m at very long times (by the equipartition theorem [4]), given which we get for the variance of the mean-square velocity We adopt a similar procedure to compute the meansquare displacement of the Brownian particle. For this, we consider which on further simplification gives us We note that the stochastic component of the force is uncorrelated with the position of the Brownian particle at all times, i.e., We consider an equilibrium state in which the meansquare velocity of the Brownian particle assumes the value / B k T m . Integrating between appropriate limits thus gives us for the mean-square displacement  Einstein diffusion constant is given as We see that this plot conforms well with Eq. (13), i.e., the mean-square displacement of the particle assumes an asymptotic value after a sufficiently long time span.

Derivation of an approximate form of the dissipation function for steady-state transient trajectories and arbitrary power-law trapping potentials
For a classical Newtonian system, it is customary to describe the system's state by an arbitrary point in phase space, which consists of the coordinates q and momenta p of all molecules and/or particles that  (1)], the reversibility of a system can be defined as the ratio of the probabilities of observing time-reversed trajectories and their corresponding time-forward trajectories. It is however a daunting task to deduce closed-form expressions for arbitrary nonequilibrium steady-state distributions in phase space [1]. We instead evolve the system (and in turn, the transient trajectories) from an equilibrium steady-state distribution, for which the dynamics is well known, i.e., we start the temporal evolution of the system at time 0 t = .
We now consider the case of a colloidal system localized in an arbitrary power-law trap which is translating at a constant acceleration in one dimension. At a particular instant t , we denote the coordinate and momenta of the colloidal particle as * t x  and t p  (we consider the motion along the xaxis), respectively. This particle is in a viscous medium and the arrangement is such that the force due to the power-law trap does not affect the neighboring solvent molecules and they all remain stationary relative to the moving colloidal particle. Thus, a simplified version of a colloid is being considered in this context. We consider constant temperature conditions, given which it is possible to apply the canonical ensemble to this system. The temperature (constant) is denoted as T . The Hamiltonian of the system can be expressed as a sum of the kinetic and potential energies of the system that arise due to intermolecular interactions and the potential energy of the colloidal particle that is trapped/confined in the moving trap. We consider very small displacements of the colloidal particle from its mean equilibrium position (i.e., the focal point of the trap) such that it always remains bound to this potential well. The Hamiltonian of the system for the time-forward trajectories assumes the form  Similarly, for the time-reversed trajectories, the Hamiltonian of the system assumes the form where we have considered a time-reversal in the phase space coordinates. We also take note of the fact that to preserve time-reversal symmetry, we must consider the trap center to be located at the point 2 (1/ 2) ( *) trap at at the start of the temporal evolution of a time-reversed trajectory. We note that in a canonical ensemble, the following proportionality relation is satisfied respectively. Note that the acceleration of the trap, trap a is constant throughout, and we place the observer in the center-of-mass frame of the moving colloidal particle/trap, given which the contributions of pseudo-force terms can be eliminated. We use Eq. (26) for the numerical analysis of the integral fluctuation theorem for such a system.

A. Harmonic trap:
For this, we allow the system to reach a steady-state equilibrium distribution initially. This results in a Maxwellian-Boltzmann distribution for the particle velocities inside the trap before the trap starts accelerating. The optical trap thus generated has a stiffness constant of k = 3.87 x 10 -7 N/m. We consider the viscosity of the medium to be  = 2.5 Poise and the radius of the colloidal particle to be R = 9.23 m. The temperature of the system is maintained at 433 K. The friction constant for the medium is given as The stochastic component of the force has been modeled by a Gaussian random number generator in the simulation.
The whole system is set in motion for about 20 seconds starting from 0 t = , from which we sample out 5000 transient particle trajectories, each spaced at a time interval of 4 x 10 -3 seconds, i.e., a time step of t  = 4 x 10 -3 seconds has been implemented in the numerical analysis. We consider a constant acceleration of trap a = 4 x 10 -7 m/s 2 for the optical trap under consideration. The setup is such that the initial position of the colloidal particle is at a distance of 0.85 μm from the bottom of the sample cell, whereas we take the initial distance of the trap center from the bottom of the sample cell to be 0.35 μm. We  . We see that the stochastic entropy distribution assumes a Gaussian profile and conforms well with all theoretical predictions [see [5]]. The parameters considered for generating the plot have been quoted in the main text.
To check the validity of the integral fluctuation theorem, which can be expressed mathematically as [3] we sample out the positive values of the stochastic entropy (i.e., entropy generation) and plot the number ratio against time t .

B. Quartic potential well:
We consider a similar scenario in this case. No changes in the parameters have been made and we follow the exact same procedure as we did for the case of the harmonic trap, i.e., Eq. (2) has been used to numerically compute the transient Brownian trajectories over a time interval of about 20 seconds (and in turn, the positions of the optical trap center relative to the colloidal particle and the walls of the sample cell at intermediate time instances). We use Eq. (26) to solve for the stochastic entropies generated and/or lost along transient particle trajectories by making the substitution 2 n = .

Discussion
We see that the results presented herein conform well with theoretical predictions. For the case of the harmonic trap, we note that beyond 2.5 seconds, the probability of observing time-reversed trajectories practically becomes zero. A similar conclusion can be drawn for the case of the quartic potential well where we see that beyond 2 seconds, the number ratio approaches zero, i.e., one does not observe any time-reversed trajectories beyond this point. Thus, we see that the integral fluctuation theorem holds for accelerated colloidal systems, at least in the longtime limit. These results can have long-lasting implications in the realization of novel nanomachines and biological motors. Micro-sized and nano-sized machines that are primarily entropydriven can be realized in principle for a wide variety of medical and industrial applications [6]. The concept of Maxwell's demon, that closely relates to the ideas presented herein has been studied in great depth over the years and has garnered significant interest amongst members of the scientific community [7]. More recently, significant advances made in the area of stochastic thermodynamics have shown that in far-from-equilibrium systems, random fluctuations are naturally constrained by the rate of entropy production through certain thermodynamic uncertainty relations [8]. To this effect, the thermodynamics of quantum nano-machines and qubit engines has been studied extensively, the results of which serve as a positive reinforcement to the idea that power outputs for certain configurations can be enhanced within a particular regime of entropy production. Many works in the existing literature discuss and investigate in depth, the importance of thermodynamics at the molecular level, especially with regards to the design and implementation of nano-scale self-assembling systems [9]. To conclude, numerous applications of this study can be envisioned and/or applied to thermodynamic systems of interest at the microscopic level.

Conclusion
A detailed mathematical analysis of the integral fluctuation theorem in the context of an accelerated colloidal system has been presented. An approximate closed-form expression for the dissipation function has been derived in the context of a colloidal system, in which a colloidal particle is bound in a potential well and is made to accelerate uniformly along one dimension, while undergoing Brownian motion inside the trap. We note that expression [see Eq. (26)] holds true for any power-law trap, hence can be generalized and analyzed beyond what has been presented in this paper. Two cases, one corresponding to a harmonic trap and the other corresponding to a quartic potential well have been considered and numerically analyzed. The results thus obtained show that the integral fluctuation theorem holds true for the form of the dissipation function deduced herein in the long-time limit. An exact, closed form generic expression for the dissipation function can also be derived using stochastic dynamics [1], although this has not been covered in this paper. initial distance being 0.35 m from the bottom of the sample cell. The parameters used in the simulation have been quoted in the main text [see section IV]. Note that the stochastic component of the total force acting on the colloidal particle was modeled using a Gaussian random number generator (with values distributed between 0 and 1). The dynamics of this accelerating system was analyzed over a timescale of 20 seconds. A total of 5000 transient particle trajectories were sampled out and studied (each being computed over a timescale of 4 ms). Eq. (26) was used to compute the stochastic entropy over each transient trajectory (note that we substitute 1 n = for the case of the harmonic trap and 2 n = for the case of the quartic potential well) and a histogram distribution of the same was generated. The computation of all transient trajectories was carried out by using Eq. (2). The results obtained conform well with theoretical predictions, as has been demonstrated in section IV. Thereafter, positive values of t  (these correspond to entropy generation) were sampled out (in accordance with Eq. (27)) and the number ratio (defined as the ratio of the probability of observing time-reversed trajectories to the probability of observing time-forward trajectories) was computed and plotted against time. As expected, it was noted that the number ratio decays exponentially with time [see Fig. 3 and Fig.  5].

Data availability
The data that supports the plots within this paper are available from the author upon reasonable request.