Entropy symmetry breaking

By a synthesis of the partition function for a supercooled liquid at some hctive temperature in the inherent structure formalism [ 1,4] with the configurational entropy obtained by restricting mrepiica replicas to be in the same state [161], Mohanty has uncovered a relationship between Parisi s replica symmetry-breaking parameter mrepiica(T), and the Narayanaswamy-Moynihan non-linear parameter x, a parameter that provides a metric on the deviation of the glassforming system from equilibrium [162] ... 

Here, Cv h(T) and Svlh(T) are the vibrational contributions to the heat capacity and the entropy, respectively. Note that the slope of the replica symmetry-breaking parameter with respect to temperature is not unity as predicted by one-step replica symmetry breaking. Rather, the slope is governed by three factors the Narayanaswamy-Moynihan nonlinearity parameter x, the Kauzmann temperature, and the ratio of the Kauzmann temperature to the glass transition temperature. 

The 19th century left us with a conflicting heritage. Classical mechanics and even quantum mechanics and relativity are time-symmetrical theories. The past and the future play the same role in them. On the other hand, thermodynamics introduces entropy, and entropy is associated to the arrow of time. So we have two descriptions of nature. Simplifying somewhat, we may say that the first emphasizes being and the second becoming. This leads to many questions. What is the role of entropy and of distance to equilibrium in nature And a second question is, how does the time-symmetry breaking of entropy relate to the laws of physics ... 

In a physically confined environment, interfacial interactions, symmetry breaking, structural frustration, and confinement-induced entropy loss can play dominant roles in determining molecular organization. Wu[295] studied the confined assembly of silica-copolymer composite mesostructures within cylindrical nanochannels of porous anodic... 

For this equality to hold, we would have to imagine that atoms within the same molecule were uncorrelated. The different atoms are, however, bonded and the configurational constraints imposed by the bonding reduce the mixing entropy from that estimated with the right-hand side of (27). Further, for the physical reasons discussed in the preceding section, we require an accurate treatment of the entropy contribution to understand transformation of phases, spontaneous assemblies and many other symmetry breaking phenomena. 

Recently, new interesting phenomena that control the mode of packing of polymers have been found, and it has been shown that the basic principles of polymer crystallography are, in some cases, violated. In particular, (1) an atactic polymer can crystallize this is, for instance, the case of polyacrylonitrile [107] (2) in a crystalline polymer the chains can be nonparallel for instance, the structure of the yform of iPP is characterized by the packing of nearly perpendicular chains [108, 109] (3) the principle of entropy-driven phase formation may be violated and the high local symmetry of the chains is lost in the limit-ordered crystkhne lattice of polymers (symmetry breaking). 

Domain I represents the domain wherein the thermodynamic solution is stable. In domain II, with the parameters noted in Fig. 1, the thermodynamic branch has become unstable owing to fluctuations in the chemical composition of the system. Beyond the thermodynamic threshold (transition point), fluctuations increase uniformly in this domain (II), eventually resulting in a new steady state which corresponds to regular spatial distributions of X and Y (Fig. 2). This state represents a low entropy, dissipative structure localized in space and whose "natural" boundaries are determined by the system itself. The spatial localization of the resultant dissipative structure demonstrates the symmetry-breaking nature of the instability. It appears that the form which the dissipative structure takes depends on the type of initial perturbation thus, the system possesses a primitive "memory effect" since the initial perturbation determines the form of the dissipative structure established. 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

Figure 3 depicts the spectmm of Lyapunov exponents in a hard-sphere system. The area below the positive Lyapunov exponent gives the value of the Kolmogorov-Sinai entropy per unit time. The positive Lyapunov exponents show that the typical trajectories are dynamically unstable. There are as many phase-space directions in which a perturbation can amplify as there are positive Lyapunov exponents. All these unstable directions are mapped onto corresponding stable directions by the time-reversal symmetry. However, the unstable phase-space directions are physically distinct from the stable ones. Therefore, systems with positive Lyapunov exponents are especially propitious for the spontaneous breaking of the time-reversal symmetry, as shown below. 

Boltzmann s W-function is not monotonic after we perform a velocity inversion of every particle—that is, if we perform time inversion. In contrast, our -function is always monotonic as long as the system is isolated. When a velocity inversion is performed, the 7f-function jumps discontinuously due to the flow of entropy from outside. After this, the 7f-function continues its monotonic decrease [10]. Our -function breaks time symmetry, because At itself breaks time symmetry. 

An alternative way to clarify the nature of this state is to test its stability with respect to a metal-insulator transition. This has received a lot of theoretical attention recently. The JT singlet ground state makes these compounds free from the tendency towards a magnetic instability observed in so many Mott insulators. In fact, their ground state does not break any symmetry and Capone et al. explained [43] that it then has a zero entropy, which makes a direct connection with a metal impossible (it would violate the Luttinger theorem). These authors predict that the only way to go from the insulator to the metal would be through an exotic superconducting phase or a first-order transition. 

This reaction is quite unlikely because the simultaneous breaking and formation of two bonds should be accompanied by a very large negative activation entropy. Moreover, the reaction, in which two electron pairs participate, requires inversion of configuration according to the Woodward-Hoffman rules of the preservation of orbital symmetry 44). The inversion would require an even number of electron pairs (parity rules45 ), and this is impossible for steric reasons ... 

---