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ABSTRACT Chemical exchange line broadening is an important phenomenon in nuclear magnetic resonance (NMR) spectroscopy, in which a nuclear spin experiences more than one magnetic environment as a result of chemical or conformational changes of a molecule. The dynamic process of chemical exchange strongly affects the sensitivity and resolution of NMR experiments and increasingly provides a powerful probe of the interconversion between chemical and conformational states of proteins, nucleic acids, and other biologic macromolecules. A simple and often used theoretic description of chemical exchange in NMR spectroscopy is based on an idealized 2-state jump model (the random phase or telegraph signal). However, chemical exchange can also be represented as a barrier crossing event that can be modeled by using chemical reaction rate theory. The timescale of crossing is determined by the barrier height, the temperature, and the dissipation modeled as collisional or frictional damping. This tutorial explores the connection between the NMR theory of chemical exchange line broadening and strong collision models for chemical kinetics in statistical mechanics. Theoretic modeling and numeric simulation are used to map the rate of barrier crossing dynamics of a particle on a potential energy surface to the chemical exchange relaxation rate constant. By developing explicit models for the exchange dynamics, the tutorial aims to elucidate the underlying dynamical processes that give rise to the rich phenomenology of chemical exchange observed in NMR spectroscopy. Software for generating and analyzing the numeric simulations is provided in the form of Python and Fortran source codes.
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Nonequilibrium statistical mechanics of money/energy exchange models
Abstract Many-body dynamical models in which Boltzmann statistics can be derived directly from the underlying dynamical laws without invoking the fundamental postulates of statistical mechanics are scarce. Interestingly, one such model is found in econophysics and in chemistry classrooms: the money game, in which players exchange money randomly in a process that resembles elastic intermolecular collisions in a gas, giving rise to the Boltzmann distribution of money owned by each player. Although this model offers a pedagogical example that demonstrates the origins of Boltzmann statistics, such demonstrations usually rely on computer simulations. In fact, a proof of the exponential steady-state distribution in this model has only become available in recent years. Here, we study this random money/energy exchange model and its extensions using a simple mean-field-type approach that examines the properties of the one-dimensional random walk performed by one of its participants. We give a simple derivation of the Boltzmann steady-state distribution in this model. Breaking the time-reversal symmetry of the game by modifying its rules results in non-Boltzmann steady-state statistics. In particular, introducing ‘unfair’ exchange rules in which a poorer player is more likely to give money to a richer player than to receive money from that richer player, results in an analytically provable Pareto-type power-law distribution of the money in the limit where the number of players is infinite, with a finite fraction of players in the ‘ground state’ (i.e. with zero money). For a finite number of players, however, the game may give rise to a bimodal distribution of money and to bistable dynamics, in which a participant’s wealth jumps between poor and rich states. The latter corresponds to a scenario where the player accumulates nearly all the available money in the game. The time evolution of a player’s wealth in this case can be thought of as a ‘chemical reaction’, where a transition between ‘reactants’ (rich state) and ‘products’ (poor state) involves crossing a large free energy barrier. We thus analyze the trajectories generated from the game using ideas from the theory of transition paths and highlight non-Markovian effects in the barrier crossing dynamics.
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- Award ID(s):
- 1955552
- PAR ID:
- 10498356
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Journal of Physics A: Mathematical and Theoretical
- Volume:
- 57
- Issue:
- 15
- ISSN:
- 1751-8113
- Format(s):
- Medium: X Size: Article No. 155003
- Size(s):
- Article No. 155003
- Sponsoring Org:
- National Science Foundation
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