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Register a new userStatistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle
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Certain fluctuations in particle number at fixed total energy lead exactly to a cut-power law distribution in the one-particle energy, via the induced fluctuations in the phase-space volume ratio. The temperature parameter is expressed automatically by an equipartition relation, while the q-parameter is related to the scaled variance and to the expectation value of the particle number. For the binomial distribution q is smaller, for the negative binomial q is larger than one. These results also represent an approximation for general particle number distributions in the reservoir up to second order in the canonical expansion. For general systems the average phase-space volume ratio expanded to second order delivers a q parameter related to the heat capacity and to the variance of the temperature. However, q differing from one leads to non-additivity of the Boltzmann-Gibbs entropy. We demonstrate that a deformed entropy, K(S), can be constructed and used for demanding additivity. This requirement leads to a second order differential equation for K(S). Finally, the generalized q-entropy formula contains the Tsallis, Renyi and Boltzmann-Gibbs-Shannon expressions as particular cases. For diverging temperature variance we obtain a novel entropy formula.
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LHC ALICE data are interpreted in terms of statistical power-law tailed pT spectra. As explanation we derive such statistical distributions for particular particle number fluctuation patterns in a finite heat bath exactly, and for general thermodynamical systems in the subleading canonical expansion approximately. Our general result, $q = 1 - 1/C + Delta T^2 / T^2$, demonstrates how the heat capacity and the temperature fluctuation effects compete, and cancel only in the standard Gaussian approximation.
The normalized probability density function (PDF) of global measures of a large class of highly correlated systems has previously been demonstrated to fall on a single non-Gaussian universal curve. We derive the functional form of the global PDF in terms of the source PDF of the individual events in the system. A single parameter distinguishes the global PDF and is related to the exponent of the source PDF. When normalized, the global PDF is shown to be insensitive to this parameter and importantly we obtain the previously demonstrated universality from an uncorrelated Gaussian source PDF. The second and third moments of the global PDF are more sensitive, providing a powerful tool to probe the degree of complexity of physical systems.
We study work extraction processes mediated by finite-time interactions with an ambient bath -- emph{partial thermalizations} -- as continuous time Markov processes for two-level systems. Such a stochastic process results in fluctuations in the amount of work that can be extracted and is characterized by the rate at which the system parameters are driven in addition to the rate of thermalization with the bath. We analyze the distribution of work for the case where the energy gap of a two-level system is driven at a constant rate. We derive analytic expressions for average work and lower bound for the variance of work showing that such processes cannot be fluctuation-free in general. We also observe that an upper bound for the Monte Carlo estimate of the variance of work can be obtained using Jarzynskis fluctuation-dissipation relation for systems initially in equilibrium. Finally, we analyse work extraction cycles by modifying the Carnot cycle, incorporating processes involving partial thermalizations and obtain efficiency at maximum power for such finite-time work extraction cycles under different sets of constraints.
The Hawkes self-excited point process provides an efficient representation of the bursty intermittent dynamics of many physical, biological, geological and economic systems. By expressing the probability for the next event per unit time (called intensity), say of an earthquake, as a sum over all past events of (possibly) long-memory kernels, the Hawkes model is non-Markovian. By mapping the Hawkes model onto stochastic partial differential equations that are Markovian, we develop a field theoretical approach in terms of probability density functionals. Solving the steady-state equations, we predict a power law scaling of the probability density function (PDF) of the intensities close to the critical point $n=1$ of the Hawkes process, with a non-universal exponent, function of the background intensity $ u_0$ of the Hawkes intensity, the average time scale of the memory kernel and the branching ratio $n$. Our theoretical predictions are confirmed by numerical simulations.
A proof of the relativistic $H$-theorem by including nonextensive effects is given. As it happens in the nonrelativistic limit, the molecular chaos hypothesis advanced by Boltzmann does not remain valid, and the second law of thermodynamics combined with a duality transformation implies that the q-parameter lies on the interval [0,2]. It is also proved that the collisional equilibrium states (null entropy source term) are described by the relativistic $q$-power law extension of the exponential Juttner distribution which reduces, in the nonrelativistic domain, to the Tsallis power law function. As a simple illustration of the basic approach, we derive the relativistic nonextensive equilibrium distribution for a dilute charged gas under the action of an electromagnetic field $F^{{mu u}}$. Such results reduce to the standard ones in the extensive limit, thereby showing that the nonextensive entropic framework can be harmonized with the space-time ideas contained in the special relativity theory.
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