Abstract
Any real physical process that produces entropy, dissipates energy as heat, or generates mechanical work must do so on a finite timescale. Recently derived thermodynamic speed limits place bounds on these observables using intrinsic timescales of the process. Here, we derive relationships for the thermodynamic speeds for any composite stochastic observable in terms of the timescales of its individual components. From these speed limits, we find bounds on thermal efficiency of stochastic processes exchanging energy as heat and work and bound the rate of entropy change in a system with entropy production and flow. Using the time set by an external clock, we find bounds on the first time to reach any value for the entropy production. As an illustration, we compute these bounds for Brownian particles diffusing in space subject to a constant-temperature heat bath and a time-dependent external force.
Funding source: National Science Foundation
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Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: This material is based upon work supported by the National Science Foundation under Grant No. 1856250.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
Appendix A: Mapping a Langevin process to master equation
In this Appendix, we derive the mapping between Langevin dynamics with a time-dependent force term, and its corresponding master equation with time-dependent transition rates. This mapping was introduced in [40]. Our starting point is the Langevin process in Eq. (3). To simulate realizations of this process, one could use the standard Euler–Mayurama method [52] (e.g., with dt = 0.01). Instead, we derive Eq. (6) by letting W y,y+Δ(t) = W L(y + Δ) and W y,y−Δ(t) = W R(y − Δ) be left and right jump probabilities, respectively. We note that all the probabilities are time-dependent, but suppress this notation. From conservation of probability, W L(y) + W R(y) ≡ 1. With these definitions, we re-write Eq. (6) as:
To compare the discrete-space process to Langevin dynamics, for small Δ we expand W L(y + Δ), W R(y − Δ) as a Taylor series, up to order ∝Δ, and P(y ±Δ) up to ∝Δ2 (we verified that keeping term of order ∝Δ2 in the jump probabilities does not change the results below). We get
Rearranging Eq. (A2), we get:
Using
Now, in the limit of Δ → 0, we can associate Eq. (A4) with the standard Fokker–Planck equation associated with Eq. (3) [52];
(and x ≈ y). Reintroducing the notation for the time dependence, we have:
We can solve the master equation numerically using Euler’s discretization for the derivative on the left and iterating over Eq. (6) from the main text (noting P(y, t)/Δ → ρ(x, t) and
![Figure A1:
Simulation results for the probability density function ρ(x, t) (red squares) obtained via integration over the Langevin process, Eq. (3). For comparison, we show P(y, t)/Δ (green squares) for similar physical parameters (provided in the main text) from integrating the master equation, Eq. (6), at t = 4π. The lattice spacing is similar to the examples in Section 2. The simulation results from these two methods agree well. The Langevin simulation was performed using the Euler–Mayurama method with dt = 0.01 and 107 particles. The master equation simulation used the Euler–Mayurama scheme with
δ
t
̂
=
δ
t
[
Δ
2
/
(
2
D
)
]
$\hat{\delta t}=\delta t[{{\Delta }}^{2}/(2D)]$
.](/document/doi/10.1515/jnet-2022-0104/asset/graphic/j_jnet-2022-0104_fig_007.jpg)
Simulation results for the probability density function ρ(x, t) (red squares) obtained via integration over the Langevin process, Eq. (3). For comparison, we show P(y, t)/Δ (green squares) for similar physical parameters (provided in the main text) from integrating the master equation, Eq. (6), at t = 4π. The lattice spacing is similar to the examples in Section 2. The simulation results from these two methods agree well. The Langevin simulation was performed using the Euler–Mayurama method with dt = 0.01 and 107 particles. The master equation simulation used the Euler–Mayurama scheme with
Appendix B: Obtaining S i and S e from the master equation
In this Appendix, we explain how to obtain the rates of entropy production S i and entropy flow S e , from the time-dependent transition rates of the master equation. The definitions and the derivation were designed to resemble those found for constant rates in Ref. [37].
Consider, for example, the simple finite lattice x − 2Δ, x − Δ, x, x + Δ, x + 2Δ. What follows extends trivially to any lattice size. The master equation is:
where we assign ρ 0 = P(x − 2Δ)/Δ, ρ 1 = P(x − Δ)/Δ, ρ 2 = P(x)/Δ and ρ 3 = P(x + Δ)/Δ, ρ 4 = P(x + 2Δ)/Δ. Importantly, in every column j; W i,i = −∑ i≠j W i ′,j , such that ∑ i W i,j ≡ 0.
In our notation,
where for every term on the diagonal; W i,i ≡ − W i,i−1 − W i,i+1, for example; W 1,1 = −W L(x − Δ) − W R(x − Δ). The transition matrix in Eq. (B2) further simplifies to (focusing now only on the right hand side, and the left 3 × 3 corner of the transition matrix, for example):
Using Eq. (B3), we can now write
The sum of every column here is zero, as it should. Note that we could write everything also with W R instead of W L, of course. Equation (B5) yields the transition rates required for S i and S e , according to Eq. (8) in the main text.
As a final step, to facilitate the numerics, here, it is useful to look at the transition matrix in the following way
This matrix shows that for any value of x, its left and right neighbor sites give the transition rate into this location from x − Δ or x + Δ, respectively. The upper and lower sites give the downward and upward exit rates (towards (x − Δ) and (x + Δ)). Using Eq. (B6), we see that in the simulation procedure, the summations in Eq. (8) can be approximated by a simple sum of nearest matrix-neighbors for any x. Additionally, we can truncate these sums entirely at finite times at very large |x|, where the value of the tails of the probability density is practically zero.
Appendix C: A standard derivation of the inverse function theorem
We briefly sketch the standard derivation of the inverse function theorem [48] used in Eq. (23). Consider the integral,
over non-zero function g(t) up to a time T(c). Choose F[T(c)] ≡ c. Now, by the Leibniz integral rule:
But, since dF[T(c)]/dc ≡ 1, we get,
the rate of change of the time T with respect to c.
Appendix D: Using numerical fitting to find the bounds on
T
S
i
(
c
)
from simulation results
To obtain the bounds on the extrinsic time
The absolute temporal rates of the Helmholtz free energy and work, obtained from simulations results, used to bound the minimal time
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