It is known that one can derive the Jarzynski equality in a stochastic process of a classical system by assuming the local detailed balance. We study how the equality is modified in the linear feedback system. There, measurement is performed on the state following a linear Langevin equation, the measured variable is linear to the state variable with white sensor noise, the Kalman filter estimates the state by utilizing measured values in the past, and a linear regulator controls the state dynamics. Although a stochastic process produced by this dynamics is non-Markovian because of the feedback loop, it is known in the control theory that a Markov process for the estimation can be separated from the whole process. To the exponent in the Jarzynski equality, we find an additional term, whose average gives the mutual information between state variables and measured variables in the Markov process for the estimation. The resultant equality holds whether the gain is optimal or not.

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In the variational principle leading to the Euler equation for a perfect fluid, we can use the method of undetermined multiplier for holonomic constraints representing mass conservation and adiabatic condition. For a dissipative fluid, the latter condition is replaced by the constraint specifying how to dissipate. Noting that this constraint is nonholonomic, we can derive the balance equation of momentum for viscous and viscoelastic fluids by using a single variational principle. We can also derive the associated Hamiltonian formulation by regarding the velocity field as the input in the framework of control theory.Comment: 15 page

We calculate the drag coefficient of a rigid spherical particle in an incompressible binary fluid mixture. A weak preferential attraction is assumed between the particle surface and one of the fluid components, and the difference in the viscosity between the two components is neglected. Using the Gaussian free-energy functional and solving the hydrodynamic equation explicitly, we can show that the preferential attraction makes the drag coefficient larger as the bulk correlation length becomes longer. The dependence of the deviation from the Stokes law on the correlation length, when it is short, turns out to be much steeper than the previous estimates.

In carrying a Brownian particle by means of an external force in one dimension from a place to another in a finite time lapse, we consider controlling the force using a feedback loop to minimize an evaluation functional. The evaluation functional contains the average of the work done to the particle by the force and two terms involving control parameters. One is a term proportional to the mean squared force, while the other is a term proportional to the mean squared distance between the final particle position and the destination. Solving differential equations numerically, we calculate the average work under optimal control for various values of control parameters and find the calculated values bounded from below. The average work is also shown to be larger as we utilize less information on the state of the particle in the feedback control by decreasing the number of observed variables and by increasing the noise amplitude in the observation.

Equations for a perfect fluid can be obtained by means of the variational principle both in the Lagrangian description and in the Eulerian one. It is known that we need additional fields somehow to describe a rotational isentropic flow in the latter description. We give a simple explanation for these fields; they are introduced to fix both ends of a pathline in the variational calculus. This restriction is imposed in the former description, and should be imposed in the latter description. It is also shown that we can derive a canonical Hamiltonian formulation for a perfect fluid by regarding the velocity field as the input in the framework of control theory.Comment: 15 page

The drag coefficient of a rigid spherical particle deviates from the Stokes law when it is put into a near-critical fluid mixture in the homogeneous phase with the critical composition. The deviation (∆γ d ) is experimentally shown to depend approximately linearly on the correlation length far from the particle (ξ ∞ ), and is suggested to be caused by the preferential attraction between one component and the particle surface. In contrast, the dependence was shown to be much steeper in the previous theoretical studies based on the Gaussian free-energy density. In the vicinity of the particle, especially when the adsorption of the preferred component makes the composition strongly off-critical, the correlation length becomes very small as compared with ξ ∞ . This spacial inhomogeneity, not considered in the previous theoretical studies, can influence the dependence of ∆γ d on ξ ∞ . To examine this possibility, we here apply the local renormalized functional theory, which was previously proposed to explain the interaction of walls immersed in a (near-)critical binary fluid mixture, describing the preferential attraction in terms of the surface field. The free-energy density in this theory, coarse-grained up to the local correlation length, has much complicated dependence on the order parameter, as compared with the Gaussian free-energy density. Still, a concise expression of the drag coefficient, which was derived in one of the previous theoretical studies, turns out to be available in the present formulation. We show that, as ξ ∞ becomes larger, the dependence of ∆γ d on ξ ∞ becomes distinctly gradual and close to the linear dependence.

We calculate the drag coefficient of a circular liquid domain in a flat fluid membrane, which is surrounded by threedimensional fluids, by using the Stokes approximation. Taking into account the difference between the membrane viscosities inside and outside the domain, we obtain an integral equation involving a parameter, which is defined as unity minus the ratio of the latter viscosity to the former. We solve the equation up to the linear order with respect to the parameter to calculate the drag coefficient numerically. Its derivative with respect to the domain viscosity, evaluated when there is no difference between the membrane viscosities, is shown to almost vanish.