Faster Uphill Relaxation in Thermodynamically Equidistant Temperature Quenches

We uncover an unforeseen asymmetry in relaxation: for a pair of thermodynamically equidistant temperature quenches, one from a lower and the other from a higher temperature, the relaxation at the ambient temperature is faster in the case of the former. We demonstrate this finding on hand of two exactly solvable many-body systems relevant in the context of single-molecule and tracer-particle dynamics. We prove that near stable minima and for all quadratic energy landscapes it is a general phenomenon that also exists in a class of non-Markovian observables probed in single-molecule and particle-tracking experiments. The asymmetry is a general feature of reversible overdamped diffusive systems with smooth single-well potentials and occurs in multiwell landscapes when quenches disturb predominantly intrawell equilibria. Our findings may be relevant for the optimization of stochastic heat engines.

Figure 1

Nonequilibrium free energy after a temperature quench $T\to T_{\mathrm{eq}}$ at time $t=0$ in units of $k_B{T}_{\mathrm{eq}}$, $\mathrm{\Delta }{F}_{\tilde{T}}=\mathcal{D}[{\mathcal{P}}_{\tilde{T}}(t=0^{+})||{\mathcal{P}}_1^{\mathrm{eq}}]$ [see Eq. (3)], as a function of the relative prequench temperature $\tilde{T}=T/T_{\mathrm{eq}}$ (note the logarithmic scale); (a) refers to the end-to-end distance of a Gaussian chain with 100 beads and (b) to the 7th in a single file of 10 particles in a linear potential with slope 10 confined to a unit box. The blue and red points depict a pair of thermodynamically equidistant temperature quenches, ${\tilde{T}}^{-}$ and ${\tilde{T}}^{+}$, with corresponding excess potential energies $⟨\mathrm{\Delta }U{⟩}_{{\tilde{T}}^{\pm }}\equiv ⟨U(0^{+}){⟩}_{{\tilde{T}}^{\pm }}-⟨U{⟩}_1$.

Figure 2

$\mathcal{D}[{\mathcal{P}}_{\tilde{T}}(t)||{\mathcal{P}}_1^{\mathrm{eq}}]$ (full lines) for the Gaussian chain [(a),(b)] and single file with 10 particles in a linear potential with slope $g=10$ [(s),(d)]. (a) refers to the entire chain of 100 beads [Eq. (4)] and (b) to the end-to-end distance [Eq. (5)] for equidistant quenches from ${\tilde{T}}^{-}=0.24$ (blue) and ${\tilde{T}}^{+}=2.64$ (red); (c) stands for the full single file for equidistant quenches from ${\tilde{T}}^{-}=0.61$ (blue) and ${\tilde{T}}^{+}=1.52$ (red); (d) the 7th particle for equidistant quenches from ${\tilde{T}}^{-}=0.54$ (blue) and ${\tilde{T}}^{+}=1.69$. The circles refer to $⟨\mathrm{\Delta }U(t){⟩}_{{\tilde{T}}^{\pm }}$ and $⟨\mathrm{\Delta }\mathcal{U}(t){⟩}_{{\tilde{T}}^{\pm }}$ in (a) and (c), and (b) and (d), respectively, and triangles denote $\mathrm{\Delta }{S}_{{\tilde{T}}^{\pm }}(t)$ and $\mathrm{\Delta }{\mathcal{S}}_{\tilde{T}}(t)$. Note the second axes for $⟨\mathrm{\Delta }U(t){⟩}_{{\tilde{T}}^{\pm }},⟨\mathrm{\Delta }\mathcal{U}(t){⟩}_{{\tilde{T}}^{\pm }}$ and $\mathrm{\Delta }{S}_{{\tilde{T}}^{\pm }}(t),\mathrm{\Delta }{\mathcal{S}}_{\tilde{T}}(t)$. Note that ${\mathcal{S}}_{\tilde{T}}(\infty )={\mathcal{S}}_1=-⟨\mathcal{U}{⟩}_1$.

Figure 3

(a) Integrand of Eq. (3) at $t=0.1$ for a one-dimensional OUp (full line) for uphill (blue) and downhill (red) relaxation with the positive $A_{+}$ and negative $A_{-}$ area under the curve. Inset: The corresponding ${\mathcal{D}}_{\tilde{T}}^M(t)$. (b), (c) Decomposition of ${\partial }_t{P}_{\tilde{T}}(x,t)$ into diffusive ${\partial }_x^2{P}_{\tilde{T}}$ (circles) and advective ${\partial }_{x}x{P}_{\tilde{T}}$ (triangles) contribution for uphill (b) and downhill (c) relaxation. Dashed lines correspond to free diffusion evolving from the same initial condition.