Langevin formula

Van der Holst JJM, van Cost FWA, Coehoom R, Bobbert PA (2009) Electron-hole recombination in disordered organic semiconductors validity of the Langevin formula. Phys Rev B 80 235202... 

The success of worm-like chain model for dsDNA made this model and formula (7.42) fashionable among the scientists (somewhat surprising fact of life is that there is such thing as fashion in science ) nowadays formula (7.42) is often used to fit the data for which it is not at all the most appropriate. We sincerely hope that Langevin formula (7.41) and its underlying freely-jointed model, as well as rotational isomers and other models will soon regain their rights and will be used where appropriate. 

This value may be compared with that found from the Langevin formula, see Eq. 7.21. From the comparison, the relaxation time for molecular dipoles is found ... 

The Debye-Langevin formula is applicable to polar dielectrics at definite restrictions. It achieves good fulfillment for gases and vapors at low pressure, and for highly dissolved solutions of polar liquids in nonpolar solvents. This formula is of great importance in the interpretation of molecular structures. 

We recently received a preprint from Dellago et al. [9] that proposed an algorithm for path sampling, which is based on the Langevin equation (and is therefore in the spirit of approach (A) [8]). They further derive formulas to compute rate constants that are based on correlation functions. Their method of computing rate constants is an alternative approach to the formula for the state conditional probability derived in the present manuscript. 

Laguerre polynomials 3-D. /-space 264 eigenfunctions 119, 262-3 Raman and CARS spectra 120 Landau-Teller formula 159 Langevin model equation 32... 

The last two results are rather similar to the quadratic forms given by Fox and Uhlenbeck for the transition probability for a stationary Gaussian-Markov process, their Eqs. (20) and (22) [82]. Although they did not identify the parity relationships of the matrices or obtain their time dependence explicitly, the Langevin equation that emerges from their analysis and the Doob formula, their Eq. (25), is essentially equivalent to the most likely terminal position in the intermediate regime obtained next. 

Additional evidence on electron-cloud radii is given by diamagnetic susceptibility and by refractive index. For the well-known Larmor-Langevin theory of diamagnetism (11—13) gives for the molecular diamagnetic susceptibility —Xm the formula... 

In fact, the validity of Eqs. (90) and (91) is not restricted to the simple (i.e., nonretarded) Langevin model as defined by Eq. (73). These formulas can be applied in other classical descriptions of Brownian motion in which a time-dependent diffusion coefficient can be defined. This is for instance, the case in the presence of non-Ohmic dissipation, in which case the motion of the Brownian particle is described by a retarded Langevin equation (see Section V). 

Let us now come back to the specific problem of the diffusion of a particle in an out-of-equilibrium environment. In a quasi-stationary regime, the particle velocity obeys the generalized Langevin equation (22). The generalized susceptibilities of interest are the particle mobility p(co) = Xvxi03) and the generalized friction coefficient y(co) = — (l/mm)x ( ) [the latter formula deriving from the relation (170) between y(f) and Xj> (f))- The results of linear response theory as applied to the particle velocity, namely the Kubo formula (156) and the Einstein relation (159), are not valid out-of-equilibrium. The same... 

Interestingly, due to the linearity of the generalized Langevin equation (22), the same effective temperature T,eff(( ) can consistently be used in the modified Nyquist formula linking the noise spectral density C/ /- ([Pg.313]

First, consider a very dilute gas of molecules [16]. The conventional theory of the static dielectric susceptibility % of such a gas invokes the notion of polarizable molecules with permanent dipole moments that are partially aligned by the external electric field . Standard techniques of statistical thermodynamics produce the Langevin-Debye formula for x Per molecule that reads... 

In other words, the conventional Langevin-Debye formula is recovered but with the expectation value of the square of the dipole moment operator in place of the square of the permanent dipole moment. 

A similar formula was obtained by Langevin in 1905 and later, on more rigorous grounds, by Chapman, who calculated the value of the coefficient... 

We have here the well-known formulae of Born whose theory leads, in addition to the Langevin anisotropic term (170a), to a supplementary dipole term of the form (170b) more strongly dependent on temperature 03 = 1/Jfc r ). 

If the dielectric consists of Nx molecules, having mutually non-interacting electric dipole moments p, (241) reduces to the well-known Langevin-Debye formula ... 

On resorting to the Oausius-Mosotti or the Langevin-Debye formula, we obtain ... 

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