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Brownian motion. Simulation of the Brownian motion of a large particle, analogous to a dust particle, that collides with a large set of smaller particles, analogous to molecules of a gas, which move with different velocities in different random directions. Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas).
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
Reversible lane. The south end of Lions Gate Bridge in Vancouver, British Columbia. A reversible lane (or tidal flow) is a managed lane in which traffic may travel in either direction, depending on certain conditions. Typically, it is meant to improve traffic flow during rush hours, by having overhead traffic lights and lighted street signs ...
In the theory of probability for stochastic processes, the reflection principle for a Wiener process states that if the path of a Wiener process f (t) reaches a value f (s) = a at time t = s, then the subsequent path after time s has the same distribution as the reflection of the subsequent path about the value a. [1]
In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle under the influence of friction. It is named after Leonard Ornstein and George Eugene Uhlenbeck.
The original Langevin equation [1][2] describes Brownian motion, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid, Here, is the velocity of the particle, is its damping coefficient, and is its mass. The force acting on the particle is written as a sum of a viscous force proportional to ...
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics.It asserts that the phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time.
Introduction. A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set. [4][5] The set used to index the random variables is called the index set.