Dissipative dynamics of classical and quantum systems

Abstract

The organization is as follows. In Chapter II we introduce specific systems of interest, such as RBCs, microemulsions, tethered surfaces, and monolayer films of smectic-A and smectic-B systems. In Chapter III we calculate viscosities of three different kinds of dilute membrane suspensions:

tethered surfaces, red blood cells, and quasi-spherical vesicles.

We show that such systems have strongly frequency-dependent viscosities at low frequencies as a result of long lifetimes of long-wavelength internal fluctuation modes. In Chapter IV we have calculated the dynamical properties of tethered surface solutions in the concentrated regime. We have made the ansatz that the single membrane configuration in the concentrated regime is simply dense packing, so that the mass in the region of size RRR goes as RdR^dRd, which means that the length of the membrane scales as LLL. In other words, R∼LνR \sim L^\nuR∼Lν, where LLL is the linear dimension of the membrane. We have used this ansatz to calculate the dynamic structure factor, diffusion coefficient, and flow birefringence for semi-dilute and concentrated solutions. In Chapter V we have calculated fluctuation corrections to the damping coefficients of a freely suspended membrane with surface tension and have shown that these corrections are divergent at low frequencies. We also correct the previous calculation of Katz and Lebedev on these systems.

The second part of the thesis includes the work on dissipative quantum systems. We begin with a brief introduction to dissipative quantum systems. We mainly address the issue of how to legitimately incorporate dissipative forces in quantum dynamics. We discuss some of the earlier attempts made by many authors to introduce dissipation in quantum mechanics and also discuss their shortcomings. We also give some motivation of why it is interesting to introduce dissipative forces in quantum mechanics. All this is done in Chapter VI. In Chapter VII we discuss a problem of an electron in an external magnetic field with a dissipative environment. Using the equation of motion technique (operator equations in the quantum context), we calculate the position autocorrelation function of the electron for both high and low temperatures. The result of our calculation shows that at high temperatures the motion of an electron is diffusive, that is, the position autocorrelation grows linearly in time. In the low-temperature regime (in fact at zero temperature) the motion of the electron is sub-diffusive, that is, it grows logarithmically in time. In both cases (high and low temperatures), the pre-multiplying factor is reduced by the magnetic field value. In other words, the diffusion constant in both cases is suppressed. We have also calculated in the same chapter the orbital magnetic moment of the electron. We show that in the classical context (high temperatures) the orbital magnetic moment of the electron is zero, if and only if we take into account the skipping orbits. “Skipping orbits” is a dynamical way of explaining what is well known in the context of classical statistical mechanics (Bohr–Van Leeuwen theorem) and also makes it clear the role of boundaries in making the moment zero. The calculation of the magnetic moment in the quantum context (zero temperature) gives a Landau diamagnetism but only when dissipative forces are turned off. The general result is that the value of the magnetic moment is suppressed due to the dissipative interaction. This suppression of the value is attributed to the reduction of the wave packet due to the blocking of the evolution by the quasi-continuous “observation” by the bath of oscillators (we model the environment as a set of oscillators). This is the oft-quoted paradox: “A continuously watched pot never boils.” This is illustrated in Chapter VII. In Chapter VIII, we consider a two-level system with energy difference interacting with an external mode with frequency ω\omegaω. This mode in turn interacts with a bath of oscillators. Coupling of the oscillators of the bath to the mode of interest is treated effectively as quantum Langevin dynamics for the creation and annihilation operators of the mode. When the energy spacing of the two-level system is different from the mode energy, a read-down transition becomes impossible because the excess energy cannot be removed in the absence of any damping mechanism. We show that when the mode of interest is damped (due to its interaction with the environment) this down transition becomes feasible. The physical reason for this is that the mode acquires a width due to its coupling to the environment, compensating, therefore, for the energy mismatch. More interestingly, the rate of down transition has a non-monotonic dependence on the damping coefficient γ\gammaγ. When the damping coefficient is small compared to the mismatch ∣Δ−ω∣|\Delta - \omega|∣Δ−ω∣, the rate of down transition increases with increase in friction. On the other hand, when the friction is large compared to the mismatch, the rate of down transition goes down as a function of increasing friction. This is quite reminiscent of the Kramers problem (but in the quantum context). We have also discussed the possible application of our model to a Rydberg atom in a cavity and possibly to a deep impurity level in a semiconductor, where such a mismatch is possible. The above-mentioned calculations have been done at zero temperature because it is amenable to analytical calculations, but the general conclusions that are reached in this chapter are expected to be applicable to non-zero temperatures as well.