Langevin equation constant-temperature

Equation (16) represents the locked-dipole capture rate constant. The first term, which at the same time is the high-temperature limit, denotes the well-known Langevin rate constant... 

Ceriotti, M., Bussi, G., Parrinello, M. Langevin equation with colored noise for constant-temperature molecular dynamics simulations. Phys. Rev. Lett. 102, 020,601 (2009). doi 10. 1103/PhysRevLett.l02.020601... 

As can be seen from Eq. 2.14, classical theoiy predicts the capture cross section should vary inversely as the relative velocity of the coUiding pair and hence the Langevin rate constant, shonld be independent of the relative velocity and the temperature. Equation 2.15 predicts reasonably well the rate constants for ion-molecule reactions involving only ion-indnced dipole interactions. This indicates that a reaction occurs on every collision for many ion/molecule pairs there can be no activation energy for the reaction. The rate constant predicted is of the order of 1X 10 cm molecule s" . 

This is the Langevin equation which describes the degree of polarization in a sample when an electric field, E, is applied at temperature T. Experimentally, a poling temperature in the vicinity of Tg is used to maximize dipole motion. The maximum electric field which may be applied, typically 100 MV/m, is determined by the dielectric breakdown strength of the polymer. For amorphous polymers p E / kT 1, which places these systems well within the linear region of the Langevin function. The following linear equation for the remanent polarization results when the Clausius Mossotti equation is used to relate the dielectric constant to the dipole moment 41). 

It can be shown that a trajectory generated by integrating the Langevin equation of motion (with at least one non-vanishing atomic friction coefficient y,) maps (at constant volume) a canonical distribution of microstates at temperature To. 

However, it is not entirely clear what velocity should be used in (212) among the possibilities discussed in [104] are an explicit update U,(t + h) = U (t + h), an implicit update U,(t + h) = U (H- h), and a semi-implicit update U,(t + h) = [U (f + 5 A) + U (f -h h) /2. It has been pointed out [ 140] that, even for a Langevin equation with constant friction, there are deviations in the temperature for finite values of h. However, the semi-implicit scheme satisfies the FDT exactly for constant friction. Here we will consider a different model for the velocity, assuming that the hydrodynamic force is distributed over the time step. For simplicity we consider a single component of the velocity. 

For both paths (Equations 18.15 and 18.16 and Equations 18.17 and 18.18, respectively) all reactions are exothermic and are expected to proceed with a temperature-independent Langevin reaction rate constant see Ref. [49] and references therein for... 

Here, the shear modulus of the swollen hair G = (pRT/Mc) (v2 — j)/ (1 < ) 1/3(i 2M,/M) y n is the number of segments of the network chain L x) is the anti-Langevin fimction p is the dry density of fiie sample is the number average molecular weight between crosslinks of the rubbery phase M is the primary molecular weight R is the gas constant T is absolute temperature V2 is the volume fiaction of the polymer within the gel and y is the filler effect of the HS domain that exists in the rubbery phase. Equation (3) provides y. 

At finite temperature, stochastic fluctuations of the membrane due to thermal motion affect the dynamics of vesicles. Since the calculation of thermal fluctuations under flow conditions requires long times and large membrane sizes (in order to have a sufficient range of undulation wave vectors), simulations have been performed for a two-dimensional system in the stationary tank-treading state [213]. For comparison, in the limit of small deviations from a circle, Langevin-type equations of motion have been derived, which are highly nonlinear due to the constraint of constant perimeter length [213]. 

Momentum conservation requires that an equal and opposite force be applied to the fluid. Both discrete and continuous degrees of freedom are subject to Langevin noise in order to balance the frictional and viscous losses, and thereby keep the temperature constant. The algorithm can be applied to any Navier-Stokes solver, not just to LB models. For this reason, we will discuss the coupling within a (continuum) Navier-Stokes framework, with a general equation of state p p). We use the abbreviations for the viscosity tensor (46), and... 

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