Teaching the Principles of Statistical Dynamics
We describe a simple framework for teaching the principles that underlie the dynamical laws of transport: Fick's law of diffusion, Fourier's law of heat flow, the Newtonian viscosity law, and the massaction laws of chemical kinetics. In analogy with the way that the maximization of entropy over microstates leads to the Boltzmann distribution and predictions about equilibria, maximizing a quantity that E. T. Jaynes called "caliber" over all the possible microtrajectories leads to these dynamical laws. The principle of maximum caliber also leads to dynamical distribution functions that characterize the relative probabilities of different microtrajectories. A great source of recent interest in statistical dynamics has resulted from a new generation of singleparticle and singlemolecule experiments that make it possible to observe dynamics one trajectory at a time.
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 "One includes inertia (in Section 4) and the other fluctuations (in Section 5). The former is motivated by [8] [9] [10] and the latter by [11] [12] [13] [14] [15] [16]. In the inertia type lifts the extra state variables are velocities of the original state variables. "

 "The amount of physical time ∆í µí±¡ that elapses between turns is determined by the rate constant í µí± and the total number í µí± of marbles in the game. The number of marbles in box 1 í µí± 1 varies randomly at each turn of the game according to the following rule: Despite its apparent simplicity, the marble game can be used as a foundation for a conceptual framework that can be applied across the STEM disciplines as outlined below (Ghosh et al., 2006). 2 "
[Show abstract] [Hide abstract] ABSTRACT: Recently there have been multiple calls for curricular reforms to develop new pathways to the science, technology, engineering and math (STEM) disciplines. The Marble Game answers these calls by providing a conceptual framework for quantitative scientific modeling skills useful across all the STEM disciplines. The approach actively engages students in a process of directed scientific discovery. In a "Student Assessment of their Learning Gains" (SALG) survey, students identified this approach as producing "great gains" in their understanding of real world problems and scientific research. Using the marble game, students build a conceptual framework that applies directly to random molecularlevel processes in biology such as diffusion and interfacial transport. It is also isomorphic with a reversible firstorder chemical reaction providing conceptual preparation for chemical kinetics. The computational and mathematical framework can also be applied to investigate the predictions of quantitative physics models ranging from Newtonian mechanics through RLC circuits. To test this approach, students were asked to derive a novel theory of osmosis. The test results confirm that they were able to successfully apply the conceptual framework to a new situation under final exam conditions. The marble game thus provides a pathway to the STEM disciplines that includes quantitative biology concepts in the undergraduate curriculum  from the very first class. 
 "One includes inertia (in Section 4) and the other fluctuations (in Section 5). The former is motivated by8910 and the latter by111213141516. In the inertia type lifts the extra state variables are velocities of the original state variables. In the context of chemical reactions they are reaction fluxes y or alternatively reaction forces x. "
[Show abstract] [Hide abstract] ABSTRACT: Dynamics of chemical reactions, called massactionlaw dynamics, serves in this paper as a motivating example for investigating geometry of nonlinear nonequilibrium thermodynamics and for studying the ways to extend a mesoscopic dynamics to more microscopic levels. The geometry in which the physics involved is naturally expressed appears to be the contact geometry. Two extensions are discussed in detail. In one, the reaction fluxes or forces are adopted as independent state variables, the other takes into account fluctuations. All the time evolution equations arising in the paper are proven to be compatible among themselves and with equilibrium thermodynamics. A quantity closely related to the entropy production plays in the extended dynamics with fluxes and forces as well as in the corresponding fluctuating dynamics the same role that entropy plays in the original massactionlaw dynamics.