Fluctuating fields

We start with a simple example the decay of concentration fluctuations in a binary mixture which is in equilibrium. Let >C(r,f)=C(r,f) - be the concentration fluctuation field in the system where is the mean concentration. C is a conserved variable and thus satisfies a conthuiity equation ... 

The lower part shows a fluctuating field in phase with a rotating induced dipole, which is always directed with the field. 

However, the Casimir mode summation of matter fields differs from the one of fluctuating fields by the presence of a further independent scale, namely the Fermi energy (i.e., the chemical potential // at zero temperature) in addition to the geometric size and distance scales (e.g. the area A and the plate separation L). 

NMR spin relaxation is not a spontaneous process, it requires stimulation by a suitable fluctuating field to induce an appropriate spin transition to reestablish equilibrium magnetization. There are four main mechanisms for obtaining relaxation dipole-dipole (most significant relaxation mechanism for I = 1/2 nuclei), chemical shift anisotropy, spin rotation, and quadrupolar (most significant relaxation mechanism for I > 1/2 nuclei) (Claridge, 1999). 

Thus, the mean velocity field - as well as the fluctuation field u - are solenoidal. 

As it stands, the last term on the right-hand side of this expression is non-linear in the spatial derivatives and appears to add a new closure problem. However, using the fact that the fluctuation field is solenoidal,... 

Ui, the fluctuation field of a passive scalar molecular diffusion coefficient r is governed by... 

Solution of the equation for the fluctuating field presented in the coordinate space is characterized by the length scale l = / y/rj and by the time scale r =... 

In the second line (15) we remained only the terms quadratic in fluctuation fields, i.e. we assumed dn F ... 

Fluctuations dominate for T > For typical values fiq (350-F500) MeV and for Tc > (50 A- 70) MeV in the weak coupling limit from (26), (22) we estimate Tq< (0.6 A- 0.8)TC. If we took into account the suppression factor / of the mean field term oc e A /T, a decrease of the mass m due to the fluctuation contribution (cf. (11)), and the pseudo-Goldstone contribution (25), we would get still smaller value of T < (< 0.5TC). We see that fluctuations start to contribute at temperatures when one can still use approximate expressions (22), (20) valid in the low temperature limit. Thus the time (frequency) dependence of the fluctuating fields is important in case of CSC. 

In the work of Brown [5] and Kubo and Hashitsume [45] the starting equation is the Gilbert equation (3.43), in which the effective field is increased by a fluctuating field yielding the stochastic Gilbert equation. This equation can, as in the deterministic case, be cast into the Landau-Lifshitz form as... 

When the constant field is weak and the fluctuating field is comparable to or even larger than the constant field, the above decomposition becomes meaningless. There is no way of distinguishing between the adiabatic and nonadiabatic effects. In order to obtain an understanding of this rather complex situation, we have examined a stochastic model,14 extending the theory in Section II. The stochastic equation of motion of a spin in a random local field is written as... 

A further refinement of the harmonic oscillator model is possible, in which the lattice is put into contact with a heat bath at temperature T and remains in contact with the heat bath, so that the initial correlations decay not only through mutual interactions but also through random collisions with an external fluctuating field. Although it might appear that such a case would contain features of both the independent particle case and the harmonic oscillator model just analyzed, the resulting formalism is much closer to that required for the latter, and the results differ only in detail. The model to be discussed is specified by the equations of motion... 

While the nuclei 3H and 13C relax predominantly by the DD mechanism, relaxation of a quadrupole nucleus such as deuterium essentially involves fluctuating fields arising from interaction between the quadrupole moment and the electrical field gradient at the quadrupole nucleus [16]. If the molecular motion is sufficiently fast (decreasing branch of the correlation function, Fig. 3.20), the 2H spin-lattice relaxation time is inversely proportional to the square of the quadrupole coupling constant e2q Q/H of deuterium and the effective correlation time [16] ... 

We have chosen an anisotropic spin-environment coupling, oc az. This is a realistic model, e.g., for many designs of solid-state qubits, where the different components of the spin are influenced by entirely different environmental degrees of freedom [10, 15, 4]. While our analysis can be generalized to account for multiple-directional fluctuating fields [20], here we focus on unidirectional fluctuations (along the 2 axis). 

Specifically, we analyze the following problem a spin S is coupled to a controlled magnetic field B (stationary for now, but to be varied slowly in a Berry-phase experiment) and a randomly fluctuating field -X (f), which we treat as a random variable with the correlation function given by Sxit). Its dynamics is governed by the Larmor equation ... 

The symmetry of the problem can be used to analyze the structure of such a second-order contribution. The spin-rotational symmetry (about the B-field s direction) and the time-translational symmetry imply that (i) the longitudinal and transverse fluctuations, Xj and X, do not interfere and may be considered separately (ii) it is convenient to expand the transverse fluctuating field in circularly polarized harmonic modes, and the latter contribute independently. 

In this paper we have derived expressions for the environment-induced correction to the Berry phase, for a spin coupled to an environment. On one hand, we presented a simple quantum-mechanical derivation for the case when the environment is treated as a separate quantum system. On the other hand, we analyzed the case of a spin subject to a random classical field. The quantum-mechanical derivation provides a result which is insensitive to the antisymmetric part of the random-field correlations. In other words, the results for the Lamb shift and the Berry phase are insensitive to whether the different-time values of the random-field operator commute with each other or not. This observation gives rise to the expectation that for a random classical field, with the same noise power, one should obtain the same result. For the quantities at hand, our analysis outlined above involving classical randomly fluctuating fields has confirmed this expectation. 

Physics described by the model with so many parameters is very rich and the model is able particularly to treat heavy fermion systems. To study the model many approaches were suggested (see reviews [2-5]). They are successful for particular regions of the parameter space but no one is totally universal. In this paper we apply to PAM the generating functional approach (GFA) developed first by Kadanoff and Baym [6] for conventional systems and generalized for strongly correlated electron systems [7-10]. In particular it has been applied to the Hubbard model with arbitrary U in the X-operators formalism [10]. The approach makes it possible to derive equations for the electron Green s function (GF) in terms of variational derivatives with respect to fluctuating fields. 

Of cause in all expressions for Q°. 2)af) ah fluctuating fields should be put zero. 

Figure 15. Dynamic hysteresis-loop effects (a) magnetic viscosity and (b) sweep-rate dependence. The sweep-rate dependence amounts to a fluctuation-field [165] or sweep-rate correction to the static coercivity Hco.

The statistical average over the electronic degrees of freedom in Eq. [15] is equivalent, in the Drude model, to integration over the induced dipole moments pg and py. The Hamiltonian H, is quadratic in the induced dipoles, and the trace can be calculated exactly as a functional integral over the fluctuating fields pg and The resulting solute-solvent interaction energy... 

To recover the ideal case of Eq. (1.1) we would have to assume that (u ), vanishes. The analog simulation of Section III, however, will involve additive stochastic forces, which are an unavoidable characteristic of any electric circuit. It is therefore convenient to regard as a parameter the value of which will be determined so as to fit the experimental results. In the absence of the coupling with the variable Eq. (1.7) would describe the standard motion of a Brownian particle in an external potential field G(x). This potential is modulated by a fluctuating field The stochastic motion of in turn, is driven by the last equation of the set of Eq. (1.7), which is a standard Langevin equation with a white Gaussian noise defined by... 

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