24 Physics World February 1993 cies-these are the "natural frequencies" of motion of a mass tied to an imperfect new example essentially distinct, or isit the system and show up as peaks in the (nonlinear) spring and subject to viscous more naturally viewed as a close relative of spectral density. According to the theory, damping. For this problem, the oscillator is earlier examples of stochastic resonance? the key signature is how this peak depends also subject to a combination of periodic Though such discussions are peculiar to on the input noise strength. If the peak and random forces. see in the technical literature, the issue of grows rapidly with increasing noise, then Rather than construct a mechanical appropriate terminology has been a sticking stochastic resonance should occur. realisation, the Lancaster group built an point in this field. The problem is whether Now, the natural next step is to complete electronic circuit which obeys the same this is truly a resonance phenomenon as the picture by drawing the connection equation of motion. On the one hand, it taught in introductory physics classes, in between these considerations and the was a straightforward matter to measure the which forcing at just the right frequency "old" stochastic resonance associated with spectral density function; however, for this resultsin a large energy transfer,as the bistable systems everyone else had equation they were also able to calculate the depicted in the famous film footage of the studied. This could be done, but the function analytically, and so test the Tacoma Narrows Bridge collapse. In pre- Lancaster group noticed something inter- predictions of their theory in detail. The vious studies the connection was accepted esting: there was nothing in the analysis that experiments showed that hallmark signature as relatively weak, but the name was already required bistability. They reasoned that of stochastic resonance: as the noise established, so thereit was. But in an bistability wasn't absolutely necessary, and intensity was gradually increased from a interesting twist, the Lancaster group points that stochastic resonance could in principle low level, the response at the signalout that their "new" example istruly occur ina monostable system. Andfrequency first improved, then passed related to classic resonance. Nevertheless, furthermore, they already knew of "a through a maximum before falling off at they decide that the similarities with the promising monostable candidate" which large values of the noise input. The data "old" examples outweigh the differences, fulfilled the necessary requirements accord- were found to be in close agreement with since the key fundamental property is ing to linear response theory. This was a the theoretical predictions. retained: the addition of random noise well worn workhorse known as the damped . Rnally, the Lancaster group addresses can improve the sensitivity to a periodic Duffing equation, which describes the the name stochastic resonance itself. Is their stimulus. 0 Order through disorder: entropy strikes back From Daan Frenkel in the FOM Institute for crystal ordered because itsconstituent the experts on the panel (including two Atomic and Molecular Physics, Amsterdam, The particles are constrained to be near specific other Nobel laureates)to vote on the Netherlands points in space (lattice sites). A fluid, where existence of the hard-sphere freezing transi- no such constraints apply, is called dis- tion. The vote was even and Uhlenbeck THEsecond law of thermodynamics states ordered. Our intuitive notion about order closed the discussion by saying: "I am quite that the entropy of an isolated system and disorder suggests that a system with a sure that the transition goes a little bit increases untilit reaches a maximum. given density and energy should have a against intuition:thatis why so many Entropy is traditionally associated with higher entropy in the fluid phase than in the people have difficulty with it, and surely I disorder but recent experiments, compu- crystalline phase, and that freezing would am one of those. But this transition-it still ter simulations and theories have illustrated result in a decrease of entropy. The second might be true, you know and I don't think how the disordering effect of entropy can law of thermodynamics tells us that this can one can decide by general arguments." lead to highly ordered superstructures in only happen if this decrease in entropy is Yet, in retrospect, we can understand the certain alloys. offset by a larger increase in entropy of some hard-sphere freezing as follows:a naive Statistical mechanics can be used to give other part of the Universe. This can happen picture of a solid is a model in which all the a "microscopic" interpretation of entropy if the energy of the crystalline phase is lower molecules are confined to cells centred on and the second law: in suitable units, than that of the liquid phase such that the lattice sites. Confining the molecules to entropy is simply equal to the logarithm system can release heat on freezing, thereby cells decreases entropy (we callthis of the number of (quantum) states acces- increasing the entropy of the rest of the contribution "configurational"). However, sible to the system. If we constrain a system Universe. This simple-minded description we also gain entropy because a molecule to be in a single state, its entropy will be of the thermodynamics of freezing explains has more free volume to move in this cell zero. If we release this constraint and allow why, for a long time, it was commonly than it had in the fluid (the molecules are the system to explore all other states with thought that attractive forces between more congested in a dense fluid than in a the same energy, the entropy will increase molecules are essential for crystallisation: a solid of the same density). At sufficiently until the system is equally likely to be in any crystal can form because the lowering of the high densities, the gain in entropy due to one of these states. At that point thepotential energy of the system upon this increase in free volume exceeds the loss entropy is at a maximum and the system solidification "pays for" the decrease inin configurational entropy, and entropy has reached equilibrium. Of course, in the entropy. therefore favours crystallisation. In other process, we have lost information about the It was therefore a great surprise when, in words, an increase in macroscopic order is system: initially, we knew exactly what state the 1950s, computer simulations indicated driven by an increase of microscopic it was in, but after equilibration, the system that a fluid of hard spheres could freeze. disorder. In fact, numerous ordering can be in any one of a large number of Hard spheres do not have any interactions transitions are now known to be driven by states. We have gone from an ordered initial so the potential energy of such a system is entropy alone. situation to a more disordered final situation. always zero. Hence, if crystallisation occurs, The complexity of the order that can be In this sense, entropy is a measure for the this can only mean that the entropy of the induced by entropy only became clear amount of disorder in a system. crystalline solid is higher than that of the recently when Paul Bartlett and R H Can we "see" order or disorder? Strictly fluid! Although, at present, the hard-sphere Ottewill from Bristol University and Peter speaking, the answer is no, because we freezing transition is well established and Pusey from Edinburgh University reported cannot see the number of accessible states non-controversial, this was certainly not the experiments on the crystallisation of a of a system. However, we have another, case when it was first reported. During a colloidal suspension of a mixture of more intuitive notion of order and disorder round-table discussion on the topic in 1957 "large" and "small" polymethylmethacry- based on things that we can see. We call a George Uhlenbeck, the chairman, asked late (PMMA) particles with a size ratio of Physics World February 1993 25 0.58 (Phys. Rev. Lett. (1992) 68 3801). These mesoscopic particles have a steep Fibre sensors promise repulsive interaction, not unlike hard spheres (which are not permitted to over- lap). Bartlett and co-workers observed that improved safety underground this mixed colloid would form a crystal with a so-called AB13 structure. This structure is From Bob Lieberman at Physical Optics Corpora- began in1988, has concentrated on familiar in metal alloys (e.g. NaZn13) and tion, Torrance, California, and AT&T Bell Labora- replacing the spectroscopic cell altogether consists of a simple cubic lattice of A tories, Murray Hill, New Jersey, US with an optical fibre. The key to particles. In the middle of each A-cube is a accomplishing this is a specially designed B particle surrounded by an icosahedral ACONTINUINGhazard in coal mines is the "D-fibre" (see figurela), which allows cluster(a figure with 20 faces and 12 build up of methane gas, which can form light carried by the fibre core to interact vertices) of other Bs. The full unit cell an explosive mixture in air with disastrous with its surroundings. This approach has consists of eight such sub-cells with results. Although ignition sources in mines the benefit of requiring virtually no optical neighbouring icosahedra alternating in are limited, explosions and uncontrolled alignment in the sensing region, and orientation by 90 . methane firesstill occur far too often.