Feedback and control theory are important ideas that should form part of the education of a physicist but rarely do. This tutorial essay aims to give enough of the formal elements of control theory to satisfy the experimentalist designing or running a typical physics experiment and enough to satisfy the theorist wishing to understand its broader intellectual context. The level is generally simple, although more advanced methods are also introduced. Several types of applications are discussed, as the practical uses of feedback extend far beyond the simple regulation problems where it is most often employed. Sketches are then provided of some of the broader implications and applications of control theory, especially in biology, which are topics of active research.

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Feedback for physicists: A tutorial essay on control

This review discusses a paradigm that has become of increasing importance in the theory of quantum phase transitions, namely, the coupling of the order-parameter fluctuations to other soft modes and the resulting impossibility of constructing a simple Landau-Ginzburg-Wilson theory in terms of the order parameter only. The soft modes in question are manifestations of generic scale invariance, i.e., the appearance of long-range order in whole regions in the phase diagram. The concept of generic scale invariance and its influence on critical behavior is explained using various examples, both classical and quantum mechanical. The peculiarities of quantum phase transitions are discussed, with emphasis on the fact that they are more susceptible to the effects of generic scale invariance than their classical counterparts. Explicit examples include the quantum ferromagnetic transition in metals, with or without quenched disorder; the metal-superconductor transition at zero temperature; and the quantum antiferromagnetic transition. Analogies with classical phase transitions in liquid crystals and classical fluids are pointed out, and a unifying conceptual framework is developed for all transitions that are influenced by generic scale invariance.

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How Generic Scale Invariance Influences Quantum and Classical Phase Transitions

We present a calorimetric study of the phase behavior of octylcyanobiphenyl (8CB) liquid crystal confined to a controlled-pore glass (CPG). We used CPG matrices with characteristic void diameters ranging from 400 to 20 nm. In bulk we obtain weakly first-order isotropic to nematic (I-N) phase transition and nearly continuous character of the nematic to smectic-A (N-SmA) phase transition. In all CPG matrices the I-N transition remains weakly first order, while the N-SmA one becomes progressively suppressed with decreasing CPG pore radius. With decreased pore diameters both phase transition temperatures monotonously decrease following similar trends, but increasing the stability range of the N phase. The heat-capacity response at the weakly first order I-N and continuous N-SmA phase transitions gradually approaches the tricritical-like and three-dimensional XY behavior, respectively. The main observed features were explained using a bicomponent single pore type phenomenological model.

Calorimetric study of octylcyanobiphenyl liquid crystal confined to a controlled-pore glass.

We used a form of schlieren interferometry to measure the mean-squared amplitude and temporal autocorrelation function of concentration fluctuations driven by the presence of a gradient during the free diffusion of a urea solution into water. By taking and processing sequences of images separated in time by less than the shortest correlation time of interest, we were able to simultaneously measure dynamics at a number of different wave vectors. The technique is conceptually similar to the shadowgraph method, which has been used to make similar measurements, but the schlieren method has the advantage that the transfer function is wave-vector independent rather than oscillatory.

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Use of dynamic schlieren interferometry to study fluctuations during free diffusion.

Different kind of measurements were performed on the liquid crystal nonyloxycyanobiphenyl (9OCB) to carry out a study of the molecular dynamics in the smectic A (SmA), nematic (N), and isotropic (I) phases as well as an exhaustive analysis of both the SmA-to-N and N-to-I phase transitions. For the dynamic study, broadband dielectric spectroscopy (102 to 1.8 × 109 Hz) was used. Two orientations (parallel and perpendicular) of the molecular director with regard to the probing electric field were investigated. From this study, the static dielectric permittivity was obtained in both alignments and, in addition, the molecular motions that contribute to each one were discussed. The static dielectric data together with specific heat and volumetric determinations were analyzed, proving that both phase transitions are weakly first order, displaying a nearly tricritical behavior. However, the width of metastable regions seems to be dependent on the physical magnitude, although specific heat and volumetric determina...

Critical behavior and scaling relationships at the SmA(d)-N and N-I transitions in nonyloxycyanobiphenyl (9OCB).

We have developed a new, extremely sensitive real-space technique (intensity fluctuation microscopy) to probe the order of the nematic-smectic-A (NA) transition Using this technique, we show that the NA transition in 4'-n-octyl-4-cyanobiphenyl (8CB) is clearly first order, contrary to calorimetric studies but in agreement with conclusions drawn from front-velocity measurements We characterize the strength of the discontinuity at the first-order transition by the dimensionless quantity t(0)=(T(NA)-T*)/T(*) By precisely measuring t(0), we have made the first detailed tests of predictions based on the Halperin-Lubensky-Ma (HLM) theory of fluctuation-induced, first-order phase transitions First, we explore the effect of an external magnetic field on the NA transition Although modest fields (of order 10 T) are predicted to drive the weakly first-order transition in pure 8CB second order, we observe no such effect; we establish instead that the lower bound on this critical field is approximately 30 T Likewise, we observe no effect in mixtures of 8CB with its longer chemical homolog 4'-n-decyl-4-cyanobiphenyl (10CB) Second, we examine the dependence of t(0) as a function of 8CB-10CB mixture concentration and find that the data in mixtures with small nematic temperature range are well-fit by the parameters derived by Anisimov et al based on calorimetric measurements As we increase the nematic range (by using concentrations closer to pure 8CB), the measured t(0) deviates more and more from the HLM predictions Smectic fluctuations, which are neglected in the HLM calculation, are an obvious candidate to explain such a discrepancy, but one's naive expectation is that they would reduce t(0) below the HLM levels, whereas the observed values are too large However, a recent renormalization-group calculation concludes that smectic fluctuations, surprisingly, should indeed increase t(0), explaining the observations presented here

Two experimental tests of a fluctuation-induced first-order phase transition: intensity fluctuation microscopy at the nematic-smectic-A transition.

We study theoretically the effect of an external field on the nematic–smectic-A transition close to the
tricritical point, where fluctuation effects govern the qualitative behavior of the transition. An external
field suppresses nematic-director fluctuations by making them massive. For a fluctuation-driven firstorder
transition, we show that an external field can drive the transition second order. In an appropriate
liquid-crystal system, we predict the required magnetic field to be of order 10 T. The equivalent electric
field is of order 1 V�mm.

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Ranjan Mukhopadhyay

Papers

Two experimental tests of a fluctuation-induced first-order phase transition: intensity fluctuation microscopy at the nematic-smectic-A transition.

External-Field-Induced Tricritical Point in a Fluctuation-Driven Nematic–Smectic-ATransition