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Register a new userQuantitative analysis of Clausius inequality
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In the context of driven diffusive systems, for thermodynamic transformations over a large but finite time window, we derive an expansion of the energy balance. In particular, we characterize the transformations which minimize the energy dissipation and describe the optimal correction to the quasi-static limit. Surprisingly, in the case of transformations between homogeneous equilibrium states of an ideal gas, the optimal transformation is a sequence of inhomogeneous equilibrium states.
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We extend certain basic and general concepts of thermodynamics to discrete Markov systems exchanging work and heat with reservoirs. In this framework we show that the celebrated Clausius inequality can be generalized and becomes an equality, significantly extending several recent results. We further show that achieving zero dissipation in a system implies that detailed balance obtains, and as a consequence there is zero power production. We obtain inequalities for power production under more general circumstances and show that near equilibrium obtaining maximum power production requires dissipation to be of the same order of magnitude.
We note that the social inequality, represented by the Lorenz function obtained plotting the fraction of wealth possessed by the faction of people (starting from the poorest in an economy), or the plot or function representing the citation numbers against the respective number of papers by a scientist (starting from the highest cited paper in scientometrics), captured by the corresponding inequality indices (namely the Kolkata $k$ and the Hirsch $h$ indices respectively), are given by the fixed points of these nonlinear functions. It has been shown that under extreme competitions (in the markets or in the universities), the $k$ index approaches to an universal limiting value, as the dynamics of competition progresses. We introduce and study these indices for the inequalities of (pre-failure) avalanches (obtainable from ultrasonic emissions), given by their nonlinear size distributions in the Fiber Bundle Models (FBM) of non-brittle materials. We will show how a prior knowledge of this terminal and (almost) universal value of the $k$ index (for a range of values of the Weibull modulus characterizing the disorder, and also for uniformly dispersed disorder, in the FBM) for avalanche distributions (as the failure dynamics progresses) can help predicting the point (stress) or time (for uniform increasing rate of stress) for complete failure of the bundle. This observation has also been complemented by noting a similar (but not identical) behavior of the Hirsch index ($h$), redefined for such avalanche statistics.
By generalizing the traditional concept of heat dQ and work dW to also include their time-dependent irreversible components d_{i}Q and d_{i}W allows us to express them in terms of the instantaneous internal temperature T(t) and pressure P(t), whereas the conventional form uses the constant values T_{0} and P_{0} of the medium. This results in an extremely useful formulation of non-equilibrium thermodynamics so that the first law turns into the Gibbs fundamental relation and the Clausius inequality becomes an equality ointdQ(t)/T(t)equiv0 in all cases, a quite remarkable but unexpected result. We determine the irreversible components d_{i}Qequivd_{i}W and discuss how they can be determined to obtain the generalized dW(t) and dQ(t).
Quantitative isoperimetric inequalities for anisotropic surface energies are shown where the isoperimetric deficit controls both the Fraenkel asymmetry and a measure of the oscillation of the boundary with respect to the boundary of the corresponding Wulff shape.
To make progress in understanding the issue of memory loss and history dependence in evolving complex systems, we consider the mixing rate that specifies how fast the future states become independent of the initial condition. We propose a simple measure for assessing the mixing rate that can be directly applied to experimental data observed in any metric space $X$. For a compact phase space $X subset R^M$, we prove the following statement. If the underlying dynamical system has a unique physical measure and its dynamics is strongly mixing with respect to this measure, then our method provides an upper bound of the mixing rate. We employ our method to analyze memory loss for the system of slowly sheared granular particles with a small inertial number $I$. The shear is induced by the moving walls as well as by the linear motion of the support surface that ensures approximately linear shear throughout the sample. We show that even if $I$ is kept fixed, the rate of memory loss (considered at the time scale given by the inverse shear rate) depends erratically on the shear rate. Our study suggests a presence of bifurcations at which the rate of memory loss increases with the shear rate while it decreases away from these points. We also find that the memory loss is not a smooth process. Its rate is closely related to frequency of the sudden transitions of the force network. The loss of memory, quantified by observing evolution of force networks, is found to be correlated with the loss of correlation of shear stress measured on the system scale. Thus, we have established a direct link between the evolution of force networks and macroscopic properties of the considered system.
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