mean free path

(noun)

the average distance traveled by a particle between collisions with other particles

Related Terms

Examples of mean free path in the following topics:

  • Random Walks

    • We can get an order of magnitude feeling for how much scattering will affect the radiation field emerging from a source using the concepts if the mean free path and the random walk.
    • The mean free path for scattering (or similarly for absorption) is simply the reciprocal of the scattering coefficient $\sigma_\nu$.
    • The net displacement of the photon after $N$ free paths is
    • However, on average it is as likely to go one way as the other so the mean values of all of the ${\bf r}_i$ vectors vanishes as does the sum.

  • Gas Diffusion and Effusion

    • For effusion to occur, the hole's diameter must be smaller than the molecules' mean free path (the average distance that a gas particle travels between successive collisions with other gas particles).
    • The opening of the hole must be smaller than the mean free path because otherwise, the gas could move back and forth through the hole.
    • Trace an individual molecule to see the path it takes.

  • Eddington Approximation

    • What if you are interested in the translucent upper layers of the atmosphere within a few effective mean paths of the surface but still a few mean free paths (scattering lengths) away from the surface.
    • $J$ is the mean intensity and $H$ and $K$ are proportional to the flux and the radiation pressure, respectively.
    • If you know the temperature structure of the material you can solve the equation for the mean intensity $J_\nu$ and then you know the source function $S_\nu$ explicitly and you can use the formal solution to the radiative transfer equation to get the radiation field.

  • Rosseland Approximation

    • Because stellar atmospheres (i.e. the effective mean path) are generally thin compared to the size of the star, we can assume that this region has plane parallel symmetry; that is, the properties of the material depend only on the depth from the surface $z$.
    • Let us assume that the properties of the radiation field do not change much over a mean free path so the second term is much smaller than the first; therefore;
    • However, we can elucidate some of its properties.First, the absorption and scattering coefficients are summed harmonically so regions of the spectrum that have the least absorption or scattering will dominate the energy flow.Furthermore, the harmonic sum is weighted heavily in the region where $\partial B_\nu/\partial T$ is large, near the peak of the blackbody emission.One can define a mean absorption coefficient by
    • where $\alpha_R$ is the Rosseland mean absorption coefficient.In stellar astrophysics one often uses the column density $\Sigma$ as the independent variable rather than $z$, $d\Sigma = \rho dz$.Making the substitution yields

  • Combined Scattering and Absorption

    • The chance that the free path will end in absorption is
    • $l_*$ is the typically distance between the points of creation and destruction of a photon—it is called the diffusion length, the thermalization length, or the effective mean free path.

  • Conservative Vector Fields

    • Conservative vector fields have the following property: The line integral from one point to another is independent of the choice of path connecting the two points; it is path-independent.
    • Conservative vector fields are also irrotational, meaning that (in three dimensions) they have vanishing curl.
    • A key property of a conservative vector field is that its integral along a path depends only on the endpoints of that path, not the particular route taken.
    • The above field $\mathbf{v}(x,y,z) = (\frac{−y}{x^2+y^2}, \frac{x}{x^2+y^2}, 0)$ includes a vortex at its center, meaning it is non-irrotational; it is neither conservative, nor does it have path independence.
    • Such vortex-free regions are examples of irrotational vector fields.

  • Non-Ionic Reactions

    • By this we mean that nucleophilic and electrophilic sites in reacting molecules bond to each other.
    • Furthermore, charged species such as carbocations, carbanions, conjugate acids and conjugate bases are often intermediates on the reaction path, the overall transformation taking place in two or more discrete steps.
    • Here we shall consider two other classes of organic reactions: Free-Radical Reactions & Pericyclic Reactions.
    • One type of "free-radical reaction", alkane halogenation has already been described.
    • Indeed, the study of free-radical polymerization of alkene monomers has opened the door to modern polymer chemistry.

  • Buddhism

    • Sramana, meaning “seeker,” was a tradition that began when new philosophical groups who believed in a more austere path to spiritual freedom rejected the authority of the Vedas and the Brahmins, the priests of Vedic Hinduism, around 800-600 BCE.
    • Sitting under what became known as the Bodhi Tree, Siddhartha discovered what Buddhists call the Noble Eightfold Path and attained Buddhatva, or Enlightenment, which is said to be a state of being completely free of lust (raga), hatred (dosa) and delusion (moha).
    • He taught what he called the Middle Way or Middle Path, the character of the Noble Eightfold Path.
    • The literal meaning of Nirvana in the Sanskrit language is “blowing out” or “quenching” and is the ultimate spiritual goal of Buddhism.
    • Depictions of the bodhisattva path are a popular subject in Buddhist art.

  • Free Energy Changes in Chemical Reactions

    • Conversely, if the free energy of the products exceeds that of the reactants, the reaction will not take place.
    • An important consequence of the one-way downward path of the free energy is that once it reaches its minimum possible value, net change comes to a halt.
    • where ΔG = change in Gibbs free energy, ΔH = change in enthalpy, T = absolute temperature, and ΔS = change in entropy
    • This means that there are four possibilities for the influence that temperature can have on the spontaneity of a process:
    • This means that there is a temperature defined by $T = \frac{\Delta H}{\Delta S}$ at which the reaction is at equilibrium; the reaction will only proceed spontaneously below this temperature.

  • Expressive Qualities of Line

    • There are many different types of lines, all characterized by their lengths and widths and the paths they take.
    • This type of line can follow any path, from sinuous curves to jagged or vertical lines.
    • On the other hand, a free-flowing, perhaps broken or loosely drawn contour line will evoke a feeling of freedom and lack of self restraint.