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Equilibrium Conditions for a Particle01:23

Equilibrium Conditions for a Particle

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When an object is in equilibrium, it is either at rest or moving with a constant velocity. There are two types of equilibrium: static and dynamic. Static equilibrium occurs when an object is at rest, while dynamic equilibrium occurs when an object is moving with a constant velocity. In both cases, there must be a balance of forces acting on the object.
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Diffusion is the passive movement of substances down their concentration gradients—requiring no expenditure of cellular energy. Substances, such as molecules or ions, diffuse from an area of high concentration to an area of low concentration in the cytosol or across membranes. Eventually, the concentration will even out, with the substance moving randomly but causing no net change in concentration. Such a state is called dynamic equilibrium, which is essential for maintaining overall...
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Newton's first law of motion states that a body at rest remains at rest, or if in motion, remains in motion at constant velocity, unless acted on by a net external force. It also states that there must be a cause for any change in velocity (a change in either magnitude or direction) to occur. This cause is a net external force. For example, consider what happens to an object sliding along a rough horizontal surface. The object quickly grinds to a halt, due to the net force of friction. If...
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A reversible chemical reaction represents a chemical process that proceeds in both forward (left to right) and reverse (right to left) directions. When the rates of the forward and reverse reactions are equal, the concentrations of the reactant and product species remain constant over time and the system is at equilibrium. A special double arrow is used to emphasize the reversible nature of the reaction. The relative concentrations of reactants and products in equilibrium systems vary greatly;...
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The free energy change for a process may be viewed as a measure of its driving force. A negative value for ΔG represents a driving force for the process in the forward direction, while a positive value represents a driving force for the process in the reverse direction. When ΔGrxn is zero, the forward and reverse driving forces are equal, and the process occurs in both directions at the same rate (the system is at equilibrium).
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Computing diffusivities from particle models out of equilibrium.

Peter Embacher1, Nicolas Dirr1, Johannes Zimmer2

  • 1School of Mathematics, Cardiff University, Cardiff CF24 4AG, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences
|May 10, 2018
PubMed
Summary

A novel numerical method extracts diffusivity from stochastic particle systems, applicable to out-of-equilibrium systems and experimental data. This approach leverages the gradient flow nature of large particle systems.

Keywords:
coarse-grainingfluctuation–dissipationnon-equilibrium thermodynamicstransport coefficients

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Area of Science:

  • Statistical Mechanics
  • Computational Physics
  • Nonlinear Dynamics

Background:

  • Extracting macroscopic properties like diffusivity from microscopic particle behavior is crucial.
  • Stochastic particle systems often exhibit complex out-of-equilibrium dynamics.
  • Understanding the connection between microscopic fluctuations and macroscopic transport is a key challenge.

Purpose of the Study:

  • To develop a numerical method for extracting the diffusivity of diffusion equations from stochastic particle systems.
  • To validate the method's applicability to systems undergoing arbitrary out-of-equilibrium evolutions.
  • To provide a tool for analyzing experimental particle data.

Main Methods:

  • The method relies on the principle that large particle systems formally obey stochastic partial differential equations of gradient flow type.
  • It requires the system to be in local equilibrium and exhibit Gaussian fluctuations.
  • The fluctuation-dissipation relation is a key component of the theoretical framework.

Main Results:

  • The proposed method successfully extracts diffusivity from three classic particle models: independent random walkers, zero-range process, and symmetric simple exclusion process.
  • Comparisons with analytic solutions confirm the accuracy of the numerical extraction.
  • The method demonstrates robustness for systems in arbitrary out-of-equilibrium states.

Conclusions:

  • A new numerical strategy enables the extraction of diffusivity from stochastic particle systems, even under non-equilibrium conditions.
  • The method is validated against established models, showing good agreement with analytical results.
  • This technique offers a promising approach for analyzing experimental data from particle-based systems.