Complex Langevin equation

In our case, the complex Langevin equations that simulate the Hamiltonian Eq. 11 read... 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

The complex susceptibility of a ferrofluid in a weak applied field may be written directly from Eqs. (109) and (110) and the Langevin equations (98) and (99) [taking note of Eq. (102)] using the shift theorem for one-sided Eourier transforms, Eq. (30). Thus... 

Because of the linearity of Heisenberg-Langevin equations, the squeezing does not depend on the initial complex amplitudes of optical modes. Squeezed light... 

No calculations yet have tackled this problem in all its complexity. Typically, it is assumed that the photodissociation produces some initial distribution of pairs, and then the subsequent time evolution of the unreacted pair probability is calculated. Even this more modest program has been carried out only at the diffusion and Langevin equation levels. We briefly comment on these results, since they indicate the magnitude of the solvent and velocity relaxation effects. 

The space- and time-dependent generalized Langevin equation (4) is a phenomenological equation. The exact numerical dynamics for any given simulation is not identical to the numerical solution of an STGLE. However, the numerical solution of the equations of motion of a system of hundreds or thousands of particles is in many senses a black box. One may get some numbers, but the dynamics is so complicated that there is very little useful additional information. On the other hand, because so much more is known about the solution of STGLEs it is very useful to try and map the complex dynamics onto an... 

Thus far, we have described the time-dependent nature of polymerizing environments both through stochastic [49-51] and lattice [52,53] models capable of addressing this kind of dynamics in a complex environment. The current article focuses on the former approach, but now rephrases the earlier justification of the use of the irreversible Langevin equation, iGLE, to the polymerization problem in the context of kinetic models, and specifically the chemical stochastic equation. The nonstationarity in the solvent response due to the collective polymerization of the dense solvent now appears naturally. This leads to a clear recipe for the construction of the requisite terms in the iGLE. Namely the potential of mean force and the friction kernel as described in Section 3. With these tools in hand, the iGLE is used... 

It can be checked that Eqs. (B.12 and B.15) are identical. One recognizes in Eq. (B.15) the three terms of the Langevin equation (see Ref. [21]). However, we emphasize that expression (B.12) has many advantages since it relates directly the complex poles of P/(z—L " (z)) to the various relevant microscopic and macroscopic timescales. 

Unfortunately, it is not always possible to obtain measured values of mobility or drift velocity. A number of such values are available for simple systems such as carbon monoxide and carbon dioxide, but few have been measured for complex ions. It is possible, however, to compute mobilities from the Langevin equation. Thus, Chong and Franklin, in studying the addition of carbon monoxide to methane at a pressure of about 0.5 Torr, found the following reaction ... 

If the noise term is turned off, the system is driven towards the nearest saddle point. Therefore, the same set of equations can be used to find and test mean-field solutions. The complex Langevin method was first applied to dense melts of copolymers [74], and later to mixtures of homopolymers and copolymers [80] and to diluted polymers confined in a slit under good solvent conditions [77]. Figure 2 shows examples of average density configurations (p ) for a ternary block copolymer/homopolymer system above and below the order/disorder transition. 

To summarize, the relaxation times (or eigenvalues) of a rather complex system such as a 3-D topologically-regular network end-Unked from Rouse chains were determined analytically. In fact, one can do even better it is possible to construct all of the eigenfunctions of the network analytically (which amounts to the transformation from Cartesian coordinates to normal coordinates). Briefly, to construct the normal mode transformation, see Eqs. 84 and 85, one has to combine the Langevin equations of motion of a network jimction, Eq. 80, and the boundary conditions in the network junctions, Eqs. 87 to 92. After some algebra one finds [25,66] ... 

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