Energy In Statistical Mechanics Research Articles

A phase transition induces chaos in a predator-prey ecosystem with a dynamic fitness landscape.

In many ecosystems, natural selection can occur quickly enough to influence the population dynamics and thus future selection. This suggests the importance of extending classical population dynamics models to include such eco-evolutionary processes. Here, we describe a predator-prey model in which the prey population growth depends on a prey density-dependent fitness landscape. We show that this two-species ecosystem is capable of exhibiting chaos even in the absence of external environmental variation or noise, and that the onset of chaotic dynamics is the result of the fitness landscape reversibly alternating between epochs of stabilizing and disruptive selection. We draw an analogy between the fitness function and the free energy in statistical mechanics, allowing us to use the physical theory of first-order phase transitions to understand the onset of rapid cycling in the chaotic predator-prey dynamics. We use quantitative techniques to study the relevance of our model to observational studies of complex ecosystems, finding that the evolution-driven chaotic dynamics confer community stability at the “edge of chaos” while creating a wide distribution of opportunities for speciation during epochs of disruptive selection—a potential observable signature of chaotic eco-evolutionary dynamics in experimental studies.

Quantropy

There is a well-known analogy between statistical and quantum mechanics. In statistical mechanics, Boltzmann realized that the probability for a system in thermal equilibrium to occupy a given state is proportional to \(\exp(-E/kT)\), where \(E\) is the energy of that state. In quantum mechanics, Feynman realized that the amplitude for a system to undergo a given history is proportional to \(\exp(-S/i\hbar)\), where \(S\) is the action of that history. In statistical mechanics, we can recover Boltzmann's formula by maximizing entropy subject to a constraint on the expected energy. This raises the question: what is the quantum mechanical analogue of entropy? We give a formula for this quantity, which we call ``quantropy''. We recover Feynman's formula from assuming that histories have complex amplitudes, that these amplitudes sum to one and that the amplitudes give a stationary point of quantropy subject to a constraint on the expected action. Alternatively, we can assume the amplitudes sum to one and that they give a stationary point of a quantity that we call ``free action'', which is analogous to free energy in statistical mechanics. We compute the quantropy, expected action and free action for a free particle and draw some conclusions from the results.

Statistical thermodynamics for choice models on graphs

Formalism based on equilibrium statistical thermodynamics is applied to communication networks of decision-making individuals. It is shown that in statistical ensembles for choice models, properly defined disutility plays the same role as energy in statistical mechanics. We demonstrate additivity and extensivity of disutility and build three types of equilibrium statistical ensembles: the canonical, the grand canonical and the super-canonical. Using Boltzmann probability measure one can reproduce the logit choice model. We propose using q-distributions for temperature evolution of moments of stochastic variables. The formalism is applied to networks with fixed topologies of different degrees of symmetry, for which analytic and numerical results are presented.

A Large-Deviation Principle for Random Evolution Equations

The validity of the large-deviation principle (LDP) has important applications to various areas including statistics, especially in statistical tests (see Chernoff 1952), information theory, communication networks and risk-sensitive control. It also appears in the evaluation of the free energy in statistical mechanics (see Ruelle 1969) and Varadhan's small time estimates of the density (see Varadhan 1967). For historical background on the theory of large deviations, we refer to Deuschel and Stroock (1989) or Dembo and Zeitouni (1993). Let Q = r([0, 1], Rk) denote the space of continuous Rk-valued functions endowed with the usual topology of uniform convergence and W be a standard Brownian motion on Q. Then, for e > 0, let {XE} be the diffusion process solution of the stochastic differential equation (SDE)

Nonperturbative Flow Equations in QCD

We review the formalism of the effective average action in quantum field theory which corresponds to a coarse grained free energy in statistical mechanics. The associated exact renormalization group equation and possible nonperturbative approximations for its solution are discussed. This is applied to QCD where one observes the consecutive emergence of mesonic bound states and spontaneous chiral symmetry breaking as the coarse graining scale is lowered. We finally present a study of the chiral phase transition in two flavor QCD. A precision estimate of the universal critical equation of state for the three-dimensional 0( 4) Heisenberg model is presented. We explicitly connect the 0( 4) universal behavior near the critical temperature and zero quark mass with the physics at zero temperature and a realistic pion mass. For realistic quark masses the pion correlation length near Tc turns out to be smaller than its zero temperature value. Quantum chromodynamics (QCD) describes qualitatively different physics at different length scales. At short distances the relevant degrees of freedom are quarks and gluons which can be treated perturbatively. At long distances we observe hadrons, and an essential part of the dynamics can be encoded in the masses and interactions of mesons. Any attempt to deal with this situation analytically and to predict the meson properties from the short distance physics (as functions of the strong gauge coupling o: 8 and the current quark masses mq) has to bridge the gap between two qualitatively different effective descriptions. Two basic problems have to be mastered for an extrapolation from short distance QCD to mesonic length scales: • The effective couplings change with scale. This does not only concern the running gauge coupling, but also the coefficients of non-renormalizable opera­ tors as, for example, four quark operators. Typically, these non-renormalizable terms become important in the momentum range where 0:8 is strong and deviate substantially from their perturbative values. Consider the four-point function which results after integrating out the gluons. For heavy quarks it contains the information about the shape of the heavy quark potential whereas for light quarks the complicated spectrum of light mesons and chiral symmetry breaking are encoded in it. At distance scales around 1 fm one expects that the effec­ tive action resembles very little the form of the classical QCD action which is relevant at short distances. • Not only the couplings, but even the relevant variables or degrees of freedom are different for long distance and short distance QCD. It seems forbiddingly

An extremal principle of macromolecular evolution

This paper shows that the logarithm of the mean asymptotic productivity of the Eigen model of polynucleotide replication satisfies an extremal principle which is formally identical to the minimization of the free energy in statistical mechanics. This extremal principle implies that: (a) Fluctuations in the asymptotic mean productivity due to variations in the replication and mutation rates of the polynucleotides are determined by a macroscopic parameter which is the analogue of the mean energy in statistical mechanics: (b) The logarithm of the asymptotic mean productivity is a convex function of the sequence length of the polynucleotides. This convexity property implies evolutionary stability, the analogue of thermodynamic stability in Ising models.The macromolecular replication model represents another class of dynamical systems in evolutionary biology which are thermodynamically stable, in the sense that the dynamical system possesses an asymptotic limit in which analogues of the laws of equilibrium thermodynamics hold.