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Related Concept Videos

Equilibrium Conditions for a Particle01:23

Equilibrium Conditions for a Particle

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.
To understand the concept of equilibrium, let us first consider the forces acting on an object. When different forces act on an object, they can...
Stability of Equilibrium Configuration01:23

Stability of Equilibrium Configuration

Understanding the stability of equilibrium configurations is a fundamental part of mechanical engineering. In any system, there are three distinct types of equilibrium: stable, neutral, and unstable.
A stable equilibrium occurs when a system tends to return to its original position when given a small displacement, and the potential energy is at its minimum. An example of a stable equilibrium is when a cantilever beam is fixed at one end and a weight is attached to the other end. If the weight...
First Law: Particles in Two-dimensional Equilibrium01:18

First Law: Particles in Two-dimensional Equilibrium

Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic equilibrium involves objects in motion without acceleration; but it is important to remember that these conditions are relative. For instance, an object may be at rest when viewed from one frame of reference, but that same object would appear to be in motion when viewed by someone moving at a constant velocity.
Newton's first law tells us about the...
Stability of Equilibrium Configuration: Problem Solving01:13

Stability of Equilibrium Configuration: Problem Solving

The stability of equilibrium configurations is an important concept in physics, engineering, and other related fields. In simple terms, it refers to the tendency of an object or system to return to its equilibrium position after being disturbed. The stability of an equilibrium configuration can be analyzed by considering the potential energy function of the system and examining its behavior near the equilibrium point.
Problem-solving in the context of the stability of equilibrium configuration...
First Law: Particles in One-dimensional Equilibrium01:10

First Law: Particles in One-dimensional Equilibrium

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 we...
Oscillations about an Equilibrium Position01:04

Oscillations about an Equilibrium Position

Stability is an important concept in oscillation. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable equilibrium point will cause the object to oscillate around that point. For an unstable equilibrium point, if the object is disturbed slightly, it will not return to the equilibrium point. There are three conditions for equilibrium points—stable, unstable, and half-stable. A half-stable equilibrium point is also unstable, but is named so because...

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Related Experiment Videos

Mixed quantum-classical equilibrium: Surface hopping.

J R Schmidt1, Priya V Parandekar, John C Tully

  • 1Department of Chemistry, Yale University, New Haven, Connecticut 06520, USA.

The Journal of Chemical Physics
|August 7, 2008
PubMed
Summary
This summary is machine-generated.

Fewest switches surface hopping for quantum dynamics doesn't always yield perfect Boltzmann equilibrium, but deviations are minimal. It approaches equilibrium under specific conditions, confirmed by simulations.

Related Experiment Videos

Area of Science:

  • Computational Chemistry
  • Quantum Dynamics
  • Statistical Mechanics

Background:

  • Mixed quantum-classical (MQC) dynamics are crucial for simulating complex systems.
  • The fewest switches surface hopping (FSSH) algorithm is a widely used MQC method.
  • Understanding the equilibrium properties of FSSH is essential for its accurate application.

Purpose of the Study:

  • To re-examine the equilibrium limits of the FSSH algorithm.
  • To clarify discrepancies with previous findings on FSSH equilibrium properties.
  • To provide analytical and numerical evidence for FSSH behavior.

Main Methods:

  • Analytical re-examination of FSSH equilibrium conditions.
  • Derivation of conditions for approaching Boltzmann equilibrium.
  • Numerical simulations using a two-level quantum system coupled to a classical bath.

Main Results:

  • FSSH does not generally yield exact Boltzmann equilibrium.
  • Observed deviations from Boltzmann equilibrium are typically small in practice.
  • FSSH approaches the exact equilibrium distribution under conditions of small adiabatic splitting and/or strong nonadiabatic coupling.

Conclusions:

  • The FSSH algorithm provides a practically accurate approximation to Boltzmann equilibrium.
  • The conditions under which FSSH approaches equilibrium are now better defined.
  • Numerical simulations support the analytical findings, validating the study's conclusions.