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Multiscale Modeling of Heterophase Polymerization
Published in Hugo Hernandez, Klaus Tauer, Heterophase Polymerization, 2021
The microscopic scale can be found beyond the colloidal domain but still below the thermodynamic limit ofa system. At the thermodynamic limit, the number of molecules present in the system is large enough so that the overall behavior of the system becomes deterministic instead of probabilistic. This limiting number of molecules depends on the particular system considered, as well as on the property of interest. For example, the interfacial tension of single colloidal entities, such as water nuclei in air, just after the nucleation event (consisting of much less than 100 molecules), already corresponds to the observed macroscopic value [171].
Simulation of Crystalline Nanoporous Materials and the Computation of Adsorption/Diffusion Properties
Published in T. Grant Glover, Bin Mu, Gas Adsorption in Metal-Organic Frameworks, 2018
In statistical physics, the thermodynamic limit denotes the limiting behavior of a physical system that consists of many particles (or components) as the volume V and the number N of particles tends to infinity (see Reference 133 and reference therein). A thermodynamic system is one whose size is large enough so that fluctuations are negligible. It is defined by [134] V→∞={V→∞N/V,E/Nconstant(microcanonical ensemble)N/V,Tconstant(canonical ensemble)Nt/Vt,Tconstant(Gibbs ensemble)μ,Tconstant(grand−canonicalensemble). In general, finite-size effects can be examined by iteratively increasing the system size until the properties are converged (i.e., when they no longer change as a function of volume).
Bose-Einstein condensation of photons from the thermodynamic limit to small photon numbers
Published in Journal of Modern Optics, 2018
Robert A. Nyman, Benjamin T. Walker
With a small mode spacing (or equivalently a high temperature), the threshold is deep and narrow in the sense that there is a large jump in population for a small change in total population. In the thermodynamic limit, of infinitesimal mode spacing, the threshold is infinitely sharp, and is a true phase transition. On the other hand, for large mode spacing (low temperature), the difference between below- and above-threshold populations is indistinct, and there is a wide range of population where it is not clear if the system is above or below threshold: the threshold is broad and shallow. For extremely small systems, there is just one mode with non-negligible population, so the population of that mode is equal to the total. In that case, there is no threshold in terms of average population, although there may be distinctive correlation or fluctuation behaviour.
Maximum energy dissipation to explain velocity fields in shallow reservoirs
Published in Journal of Hydraulic Research, 2018
Martijn C. Westhoff, Sébastien Erpicum, Pierre Archambeau, Michel Pirotton, Benjamin Dewals
In different fields, it has been shown that systems evolve in such a way to operate at, or close to, their thermodynamic limit, which is a physical boundary on the system that cannot be passed. One of the best examples of such a limit is the so-called Carnot limit, describing the maximum amount of work a steam engine can perform for a given temperature gradient (Carnot, 1824). Similar limits are also present in other system settings, with different forms of energy or more degrees of freedom for a system to adapt. For example, it has been shown that the yearly mean atmospheric heat transport appears to be such that the dissipative process of heat transport maximizes entropy production (Lorenz, Lunine, Withers, & McKay, 2001; Paltridge, 1979); the statistical nature of fractal river networks can be reproduced by stating that energy dissipation of flow through the river network is minimized (Hergarten, Winkler, & Birk, 2014; Howard, 1990; Rinaldo et al., 1992; Rodriguez-Iturbe, Rinaldo, Rigon, Bras, Ijjasz-Vasquez et al., 1992; Rodríguez-Iturbe, Rinaldo, Rigon, Bras, Marani et al., 1992); river meanders can be predicted by minimizing the variance of shear and the friction factor, leading to the most probable form of channel geometry (Langbein and Leopold, 1966); the maximum power principle can be used to predict vertical turbulent heat fluxes (Kleidon & Renner, 2013) or the development of preferential river flow structures at the continental scale (Kleidon, Zehe, Ehret, & Scherer, 2013); while enhanced infiltration of rainwater by preferential macropore structures is explained by the principle of maximum free energy dissipation (Zehe et al., 2013). Whilst these extremum principles appear to be contradictory at first sight, they are merely two sides of the same coin. For example, if power is performed on a system, entropy is also produced, since motion is always associated with frictional losses. In steady state systems, all power is balanced by dissipation, and hence maximizing power is equivalent to maximizing dissipation and entropy production (Kleidon, 2016).