The microscopic basis of Newton's law of cooling and its modification when the difference in temperature between the system and the surroundings is very large is discussed. When the system of interest is interacting with a small bath, the effect of the dynamical evolution of the bath variables is important to find out its dynamical feedback on the system. As in the usual system-bath approach, however, the bath is finally considered to be in thermal equilibrium and thereby provides an effective generalization of the Born-Markov master equation. It is shown that the cooling at early time is faster than that predicted by Newton's law due to the dynamical feedback of the bath.
The counterintuitive Mpemba effect, wherein a hotter system cools faster, critically lacks a general macroscopic theory. Here, starting from linear irreversible thermodynamics, we formulate a generalized Newton's cooling law,
dT/dt=−[γ0+MQ(t)](T−Tr), for a system at temperature
T relaxing in a thermal reservoir at
Tr, where the bare relaxation rate
γ0 is modified by an initial-state memory term,
Q(t)∝T(0)−Tr. Arising from the interplay between heat flux and structural evolution, the coefficient
M governs anomalous relaxation behaviors, where
M>0 (
M<0) induces the (inverse) Mpemba effect. This universal thermodynamic framework maps out phase diagram to provide general criteria for the Mpemba effect in complex systems, offering a macroscopic picture that bridges disparate microscopic approaches.
The procedure used to "do physics" in the macroscopic world is familiar: You take an object, start it off with a particular position and velocity, subject it to known forces (say gravity or friction, or both), and follow its trajectory. You then measure the dynamical properties (say position or energy) of that object at a later time and compare those measurements with the prediction using Isaac Newton's laws of motion. Newton's laws directly predict what those quantities should be at that later time, so the comparison is straightforward. However, the microscopic laws of physics, quantum mechanics, aren't so simple. The quantum concepts are so alien and counterintuitive that the language used to describe the mathematics and physics is often ambiguous and confusing. Therefore, for you to \emph{really} understand what is going on, some mathematics must be used. This is my attempt to explain some "weird facts" using a minimum of mathematics, yet eliminating as much ambiguity as possible.
Ever since the advent of quantum mechanics, it has been clear that the atoms composing matter do not obey Newton's laws. Instead, their behavior is described by the Schroedinger equation. Surprisingly though, until recently, no clear explanation was given for why everyday objects, which are merely collections of atoms, are observed to obey Newton's laws. It would seem that, if quantum mechanics explains all the properties of atoms accurately, they, too, should obey quantum mechanics. This reasoning led some scientists to believe in a distinct macroscopic, or ``big and complicated,'' world in which quantum mechanics fails and classical mechanics takes over, although there has never been experimental evidence for such a failure. Even those who insisted that Newtonian mechanics would somehow emerge from the underlying quantum mechanics as the system became increasingly macroscopic were hindered by the lack of adequate experimental and theoretical tools. In the last decade, however, this quantum-to-classical transition has become accessible to experimental study and quantitative description, and the resulting insights are the subject of this article.
With some simple examples behind us, we are now ready to list the rules of quantum mechanics: that is, the rules that we need. Unfortunately, there are more than the three laws of classical mechanics (Newton's laws) or the four laws of thermodynamics. But the rules we need can be written on one page. The remainder of the chapter illustrates the meaning of some of these rules, using simple wave functions that we have already derived. The best way to understand these rules is by practicing their application. For this reason, more end-of-chapter problems are present here than in any other chapter.