Reversibility in nonequilibrium trajectories of an optically trapped particle

The fluctuation theorem (FT) describes how a system’s thermodynamic irreversibility develops in time from a completely thermodynamically reversible system at short observation times, to a thermodynamically irreversible one at infinitely long times. In this paper, we present a general definition of the dissipation function ${\Omega }_t$, the quantitative argument in the fluctuation theorem (FT), that is a measure of a system’s irreversibility. Originally cast for deterministic systems, we demonstrate, through the example of two recent experiments, that the dissipation function can be defined for stochastic systems. While the ensemble average of ${\Omega }_t$ is positive definite irrespective of the system for which it is constructed, different expressions for ${\Omega }_t$ can arise in stochastic and deterministic systems. Moreover, within the stochastic framework, ${\Omega }_t$ is not unique. Nevertheless, each of these expressions for ${\Omega }_t$ satisfies the FT.

Figure 1

(Color online) An illustration of a set of neighboring Newtonian trajectories initiated in a volume element $\delta V_{\Gamma }\left({\mathit{\Gamma }}_0\right)$ (top tube) and the corresponding set of time reverse or antitrajectories initiated in $\delta V_{\Gamma }\left({\mathit{\Gamma }}_0^{*}\right)$ (lower tube) in coordinate, momentum $(\mathit{\Gamma }\equiv \mathbf{q},\mathbf{p})$ and time $t$ space. For every trajectory that starts at ${\mathit{\Gamma }}_0\equiv ({\mathbf{q}}_0,{\mathbf{p}}_0)$ in volume element $\delta V_{\Gamma }\left({\mathit{\Gamma }}_0\right)$ and ends at ${\mathit{\Gamma }}_t\equiv ({\mathbf{q}}_t,{\mathbf{p}}_t)$ in volume element $\delta V_{\Gamma }\left({\mathit{\Gamma }}_t\right)$ at some time $t$ later, there exists its time reversed or antitrajectory that starts at ${\mathit{\Gamma }}_0^{*}\equiv ({\mathbf{q}}_t,-{\mathbf{p}}_t)$ in element $\delta V_{\Gamma }\left({\mathit{\Gamma }}_0^{*}\right)$ and ends at ${\mathit{\Gamma }}_t^{*}\equiv ({\mathbf{q}}_0,-{\mathbf{p}}_0)$ in $\delta V_{\Gamma }\left({\mathit{\Gamma }}_t^{*}\right)$. The size of the volume elements $\delta V_{\Gamma }\left({\mathit{\Gamma }}_t\right)$ and $\delta V_{\Gamma }\left({\mathit{\Gamma }}_0^{*}\right)$ are equivalent by time reversal of the equations of motion. For thermostated, dissipative systems, there is a contraction of phase-space volume in time [i.e., $\delta V_{\Gamma }\left({\mathit{\Gamma }}_t\right)<\delta V_{\Gamma }\left({\mathit{\Gamma }}_0\right)$], represented in the figure by the tube cross section shrinking in time, and quantified by a thermodynamic entropy loss.

Figure 2

(Color online) An illustration of a subset of the stochastic trajectories that initiate at ${\mathbf{r}}_0$ and terminate at ${\mathbf{r}}_t$, denoted by $\{{\mathbf{r}}_0,{\mathbf{r}}_t\}$, and the corresponding subset of backwards trajectories $\{{\mathbf{r}}_t,{\mathbf{r}}_0\}$. Such stochastic trajectories are represented by the position of the Brownian particle(s) at time $s$, ${\mathbf{r}}_s$ from $0<s<t$; i.e., the degrees of freedom of the solvent molecules are reduced to a material parameter, viscosity, and a random fluctuating force. The stochastic Langevin equation of motion is not time reversible and consequently, it is not possible to construct an antitrajectory that is conjugate to any particular trajectory as is possible using Newtonian dynamics. Moreover, as position is not unique to any given stochastic trajectory, a bundle of trajectories is determined by both initial and terminating positions, ${\mathbf{r}}_0$ and ${\mathbf{r}}_t$, unlike deterministic trajectories where each trajectory is fully defined by only one state point ${\mathit{\Gamma }}_s$ at any time $s$ along its trajectory. While it is artificial to construct conjugate pairs of stochastic trajectories, it is possible to numerically construct a set of forward trajectories $\{{\mathbf{r}}_0,{\mathbf{r}}_t\}$ and a set of backwards trajectories $\{{\mathbf{r}}_t,{\mathbf{r}}_0\}$ whose probability distribution can be expressed analytically or constructed numerically from simulation.

Figure 3

(Color online) Distributions of $P\left({\Omega }_t\right)$ at $t=(•)2\text{ms},(•)20\text{ms (blue online)},(•)200\text{ms (red online)}$ constructed from Langevin simulation of a particle in a two-dimensional harmonic well whose strength increases from $k_0$ to $k_1$ at $t=0$. The values of the characteristic time, $\tau \equiv \xi ∕k_1$, and trapping constants have been chosen to mimic the experiment of Carberry et al. [13]: $\xi =1.05\times {10}^{-7}\text{Ns}∕\mathrm{m}$, $k_0=1.22\text{pN}∕\mu \mathrm{m}$, and the orthogonally resolved trapping constants of $k_1^x=2.9$ and $k_1^y=2.7\text{pN}∕\mu \mathrm{m}$. The simulated distributions are strikingly similar to the distributions sampled in the capture experiment [13], and within $1\%$ of an analytic formula $P({\Omega }_t=A)=\exp \text{[}A∕2-\sqrt{a_t^{-1}+(1∕4)}\mid A\mid ]∕\sqrt{a_t^2+4a_t}$ where $a_t=\left[1-\exp \left(-2t∕\tau \right)\right]{(k_0-k_1)}^2∕\left(k_0{k}_1\right)$. The inset plots $\ln \left[P\right({\Omega }_t=A)∕P({\Omega }_t=-A\left)\right]$ against ${\Omega }_t$ for these three distributions, and compares them with the prediction of the TFT (line).

Figure 4

(Color online) The ensemble average of the dissipation function $⟨{\Omega }_t⟩$ versus trajectory time $t$ from Carberry’s capture experiment (points) and predicted from Langevin dynamics, Eqs. (21, 23).

Figure 5

(Color online) Two example trajectories generated for the drag experiment using Langevin dynamics, one from a set of forward trajectories $\{{\mathbf{r}}_0,{\mathbf{r}}_t\}$ and another from a set of backward trajectories $\{{\mathbf{r}}_t,{\mathbf{r}}_0\}$. These trajectories are from conjugate trajectory sets in the $\mathbf{r}$-coordinate frame as the initial position in the forward trajectory is identical to the destination position in the backward trajectory [see dashed lines in (a)]. In (b), these same trajectories are transformed into the moving $\mathbf{x}$-coordinate frame where they are no longer members of conjugate trajectory sets. The forward trajectory initiates at ${\mathbf{x}}_0$ and terminates at ${\mathbf{x}}_t$, while the backward trajectory starts at ${\mathbf{x}}_t^{\prime }$ (where ${\mathbf{x}}_t^{\prime }\ne {\mathbf{x}}_t$) and ends at ${\mathbf{x}}_0^{\prime }\left({\mathbf{x}}_0^{\prime }\ne {\mathbf{x}}_0\right)$.

Figure 6

(Color online) Langevin simulations of the drag experiment for three different dissipation functions; ${\Omega }_t^N(•)$ (red online), ${\Omega }_t^{L,x}(•)$ (blue online), ${\Omega }_t^{L,r}(•)$ (green online). The dots represent the probability ratio, $P({\Omega }_t<0)∕P({\Omega }_t>0)$, and the lines represent the average of the exponential, $\exp (-{\Omega }_t)$, Eq. (15). The parameters for this simulation are; $k=0.5$, $\mathbf{v}=2$, $\xi =3$, where energy is in units of $k_{B}T$, length is in micrometers, and time is in seconds. All three dissipation functions obey the ITFT, but have different values at different times.