Landauer's Principle and Divergenceless Dynamical Systems

Claudia Zander, Angel Plastino, Angelo Plastino, Montserrat Casas, Sergio Curilef
2009 Entropy  
Landauer's principle is one of the pillars of the physics of information. It constitutes one of the foundations behind the idea that "information is physical". Landauer's principle establishes the smallest amount of energy that has to be dissipated when one bit of information is erased from a computing device. Here we explore an extended Landauerlike principle valid for general dynamical systems (not necessarily Hamiltonian) governed by divergenceless phase space flows. The physics of
more » ... hysics of information constitutes an active research field that has been the focus of considerable attention in recent times [1-10]. Due to seminal results generated by these research efforts the physical Entropy 2009, 11 587 reality of information is by now generally acknowledged. In this regard, the ultimate performance limits imposed by the laws of physics on any real device that processes or transmits information are starting to be understood [6, 11] . On the other hand, several theoretical developments indicate that the concept of information is essential for understanding the basic fabric of the physical world [1] [2] [3] [4] [5] [6] . Tools inspired by information-theoretical ideas, such as the maximum entropy (maxent) principle [12] [13] [14] have been successfully applied to the study of several physical scenarios. Last, but certainly not least, the ideas and methods from the physics of information lead to important points of contact between physics and biology. In fact, information processing is clearly at the very heart of biology and has been appropriately dubbed the "touchstone of life" [15] . Landauer's principle is one of the most fundamental results in the physics of information. It constituted a historical landmark in the development of the field by directly connecting information processing with conventional physical quantities [16] . Most remarkably, it played a prominent role in the final defeat of Maxwell's demon [6] . Landauer's principle states that there is a minimum amount of energy that has to be dissipated, on average, when erasing a bit of information in a computing device working at absolute temperature T . This minimum energy is equal to kT ln 2, where k is Boltzmann's constant [17] [18] [19] [20] . Landauer's principle has profound implications as it allows for novel, physically motivated derivations of several important results in classical and quantum information theory [21] . Moreover, it proved to be a powerful heuristic tool for establishing new links between, or obtaining new derivations of, fundamental aspects of thermodynamics and other areas of physics [22] . It is fair to say that most derivations of Landauer's principle can be regarded as semi-phenomenological, since they are based on a direct application of the second principle of thermodynamics. However, derivations based upon dynamical principles have also been advanced. They assume that the systems under consideration are governed by a Hamiltonian dynamics and are in thermal equilibrium, implying that they can be described by Gibb's canonical distributions. In view of the fundamental character of Landauer's principle, however, it is highly desirable to explore extensions of it applicable to systems governed by more general kinds of dynamics. These developments are inscribed within the more general program of extending the methods of statistical mechanics to non-Hamiltonian systems [23, 24] . Of special relevance is the class of dynamical systems with divergenceless phase-space flows, that include Hamiltonian systems as particular members. Divergenceless systems are characterized by the remarkable property that their dynamics preserves information. There are interesting divergenceless dynamical systems in physics, theoretical biology and other areas that are not Hamiltonian, or that have their most natural description in terms of a non-canonical set of variables. For example, the Lotka-Volterra predator-prey systems [26, 27] and the Nambu systems [28] share the vanishing divergence property. The Lotka-Volterra predator-prey systems constitute some of the most important dynamical systems considered in theoretical biology [26] . Nambu systems have been the focus of considerable research activity (see [29] [30] [31] [32] [33] and references therein). The main difference between Hamiltonian systems and Nambu systems is that, while the dynamics of a Hamiltonian system is governed by one single phase-space function (the Hamiltonian function) the dynamics of a Nambu system is governed by a set of N (N ≥ 2) such functions or "Hamiltonians" [28] . The dynamics of Nambu systems can be formulated in terms of Poisson-like brackets involving, in general, more than two functions. In the case of a Nambu system with N "Hamiltonians" the time derivative of a general phase-space function A is given by an appropriate (N + 1)-bracket de-
doi:10.3390/e11040586 fatcat:pjnpyii3nrcgfbu7wk3l574wwq