Abstract
The Mpemba effect refers to systems whose thermal relaxation time is a non-monotonic function of the initial temperature. Thus, a system that is initially hot cools to a bath temperature more quickly than the same system, initially warm. In the special case where the system dynamics can be described by a double-well potential with metastable and stable states, dynamics occurs in two stages: a fast relaxation to local equilibrium followed by a slow equilibration of populations in each coarse-grained state. We have recently observed the Mpemba effect experimentally in such a setting, for a colloidal particle immersed in water. Here, we show that this metastable Mpemba effect arises from a non-monotonic temperature dependence of the maximum amount of work that can be extracted from the local-equilibrium state at the end of Stage 1.
Highlights
A generic consequence of the second law of thermodynamics is that a system, once perturbed, will tend to relax back to thermal equilibrium
Inspired by the scenario proposed by Lu and Raz [1], we have explored the Mpemba effect in a simple, mesoscopic setting that—unlike previous work—lends itself to quantitative experiments that straightforwardly connect with theory [25]
What physical picture corresponds to the anomalous temperature dependence of the a2 coefficient? In this Brief Research Report, we offer a more physical interpretation of the Mpemba effect explored in our previous work
Summary
A generic consequence of the second law of thermodynamics is that a system, once perturbed, will tend to relax back to thermal equilibrium. Glassy systems and other complex materials may have a spectrum of exponents for mechanical and dielectric relaxation that have very long time scales and many closely spaced values that are not resolved as a sequence of exponentials Rather, they can collectively combine to approximate a power-law or even logarithmic time decay, with specific details that depend on the history of preparation [2]. The essence of Lu and Raz’s explanation is that the projection of the initial state p(x, 0)—a Gibbs-Boltzmann distribution corresponding to an initial temperature T—onto the slowest eigenfunction, a2 can be non-monotonic in T, or, equivalently, in β−1 ≡ kBT, where kB ≡ 1 (in our units) is Boltzmann’s constant Such a consequence implies a Mpemba effect because the longtime limit for the probability density function has the same form as Equation (3): p(x, t) ≈ gβb (x) + a2(β, βb) v2,βb (x) e−λ2t ,. Using marginalization and the definition of conditional probability, we can write such a local-equilibrium state as
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