Theoretical and Computational Science About a World Where Second Law of Thermodynamics Can Be Violated

Copyright: © 2014 Chelli R. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. If you ask a biologist, a chemist or a physician to enunciate the Second Law of Thermodynamics (SLT), you probably will get different answers. Besides the famous and synthetic statement “the entropy of an isolated system never decreases” [1], several other empirical formulations will be provided, such as those by Clausius, “No process is possible whose sole result is the transfer of heat from a cooler to a hotter body” [1], and by Kelvin-Planck, “No process is possible whose sole result is the absorption of heat from a reservoir and the conversion of this heat into work” [1]. Although these definitions may appear conceptually distant from each other, they represent different faces of the same physical principle, as their equivalence can be easily demonstrated [1].

If you ask a biologist, a chemist or a physician to enunciate the Second Law of Thermodynamics (SLT), you probably will get different answers. Besides the famous and synthetic statement "the entropy of an isolated system never decreases" [1], several other empirical formulations will be provided, such as those by Clausius, "No process is possible whose sole result is the transfer of heat from a cooler to a hotter body" [1], and by Kelvin-Planck, "No process is possible whose sole result is the absorption of heat from a reservoir and the conversion of this heat into work" [1]. Although these definitions may appear conceptually distant from each other, they represent different faces of the same physical principle, as their equivalence can be easily demonstrated [1].
In many other cases, answers to your question will focus on more formal statements. The Clausius-Duhem inequality is perhaps the most popular formal statement of the SLT. It applies to thermodynamic processes during which a system evolves from one equilibrium state (A) to another (B). It asserts that the integrated heat absorbed by the system, divided by the temperature at which that heat is absorbed, is bounded from above by the net change in the entropy of the system: d By "thermodynamic process", we have in mind a situation in which the system is brought into thermal contact with a sequence of heat reservoirs at different temperatures, one at a time, while one or more external parameters of the system are varied with time producing a net work. In general, the process is irreversible: it is carried out over a finite time with a finite number of reservoirs, and the system evolves from A to B through a sequence of nonequilibrium intermediate states. In such a case, the inequality strictly holds in Equation 1.
From the formal point of view nothing would seem anomalous in these answers. Now, let suppose to ask the following question: "if I put in thermal contact two iron bodies at different temperatures for a given finite time, would it be possible to observe enhancing of temperature in the hotter body?". Most people, invoking Clausius, would answer: "that's impossible!" Well, any experiment, if carefully realized, would confirm that answer. Notwithstanding, the answer would not be (entirely) correct.
Macroscopic thermodynamics is essentially based on principles or postulates that are valid on empirical basis. Till now, no observation has brought to the conclusion that heat flows spontaneously from cooler to hotter bodies: therefore Clausius' statement can be, and indeed it has been, assumed as a postulate of classical thermodynamics. However, when we move from macroscopic to microscopic world, matter could reveal aspects much more complex and bizarre than our macroscopic eyes are able to catch. These differences are dramatically evident in the outcomes of classical and quantum mechanics: from Heisenberg on, we left determinism as a natural principle, but continued to keep it for our macroscopic matters. In this respect, classical and statistical thermodynamics, as descriptions of thermodynamic events at the macroscopic and microscopic levels, respectively, may present strange and intriguing differences, and indeed they do. As I will point out, their relation resembles the one existing between classical and quantum mechanics, in the sense that the former can be safely taken as a very good approximation of the latter as macroscopic processes are considered.
Important aspects of thermodynamics emerging from a statistical microscopic analysis have been disclosed at the end of the past millennium by Christopher Jarzynski, who introduced a general nonequilibrium approach to compute free energy differences between two thermodynamic states from a series of realizations which drive the system arbitrarily far from equilibrium [2]. Specifically, the Jarzynski's approach relies on an exact relationship between the free energy difference : there is only one heat reservoir and its temperature T must equal that at which the system is initially prepared. Equation 2 has been derived in a number of ways, and confirmed in single-molecule experiments [4]. This result just represents a statistical steady statement of the SLT. More surprisingly, it contains the classical expression of the SLT which involves the free energy difference, namely W F ≥ ∆ . In fact, by exploiting the convexity of the function e x in Equation 2, i.e., e e x x ≥ , we obtain (3) which is the SLT in the statistical sense. Thus, the work averaged over all possible realizations of a process, rather than the work performed in a single realization of the process, is always greater than,

Abstract
One of the most important outcomes of thermodynamics since its first statements was formulated in 1997 by Christopher Jarzynski and expresses the second law of thermodynamics in equality form, rather than in the scholarly known (inequality) form. It also put precise quantitative conditions for violating the second law of thermodynamics, which can be realized only when dealing with microscopic nonequilibrium processes. There exist a statistical statement of it? The answer to this question was given by Jarzynski himself in [5], where he proposed an extension of Equation 2 to a process in which the system can be put into contact with several thermal baths during irreversible transformations, the first and the last bath being at the temperatures T A and T B , respectively: (4) Here, as in Equation 2, the average is estimated from an infinite number of realizations leading the system from A to B, reversibly or irreversibly does not matter. In Equation 4, ( ) This result expresses the entropy difference

SLT violations!
In conclusion, what about Clausius' and Kelvin-Planck's statements for macroscopic systems? Only a light restoration. "It is extremely improbable (practically impossible) to observe a macroscopic process whose sole result is the transfer of heat from a cooler to a hotter body" (Clausius). "It is extremely improbable (practically impossible) to observe a macroscopic process whose sole result is the absorption of heat from a reservoir and the conversion of this heat into work" (Kelvin-Planck).
Note the two additions: the sentence "It is extremely improbable" and the adjective "macroscopic". A general definition of the SLT cannot however disregard from using the cool mathematical language of Equation 4 or, if you prefer, the equivalent Equation 5.
Beyond this report, the interested Readers can find stimulating discussions in [7].