Most of the properties that we calculate for a molecular system are averages. Well known properties like temperature, pressure and density are calculated as ensemble averages, and in the real world they are treated as fixed, measurable quantities, which they generally appear to be. However all averages are obtained by summing over many numbers, and it would be very unusual (even pointless) if all the individual numbers summed had exactly the same value. Thus in practice we expect the average to show some dispersion - individual contributions are scatteerd about the mean value. In statistical thermodynamics this dispersion about the average value is known as fluctuation and it is both a subtle and important property of all physical systems.

When calculating an ensemble average (of say, pressure at fixed temperature and density), we take an instantaneous snapshot of a very large set of replicas of the system concerned and compute the average from the sum of the individual values taken from each replica. Even though each replica represents the same system at the same pressure, their individual, instantaneous values differ slightly, because the molecules that bombard the vessel surfaces to create the pressure are not in synchronisation between each replica and cannot possibly give rise to precisely the same surface forces at the same instant. Thus, with pressure, we expect some fluctuation about the mean value and indeed, similar arguments can be made for all the bulk properties of the system.

Fluctuations are of fundamental importance in statistical mechanics because they provide the means by which many physical properties of a molecular system can happen. For instance, the density of a liquid at equilibrium is a fixed, uniform quantity and we feel justified in considering the system to be isotropic - the same at all points within its bulk. Yet we know that the molecules in the system are undergoing diffusion and can easily travel throughout the bulk of the liquid. It is diffcult to imagine how this diffusion can take place if the environment each molecule is in is completely isotropic. If however we consider the density to be fluctuating minutely from the mean value at different points in the bulk, we can readily see that such fluctuations would provide a means by which the diffusion may take place. It is a surprising fact, but most of the physical properties of a bulk system are driven by fluctations, and indeed can be calculated directly from them. For this reason it is possible to view fluctations as even more fundamental than the average value.

A good example of the importance of fluctuation is provided by the Fluctuation-Dissipation theorem, which is a theorem of great power in statistical mechanics. This theorem proposes that the mechanism underpinning the response of a system to an external perturbation, is precisely the same mechanism by which equilibrium fluctuations are held close to the average bulk vaue. Thus for example, a molecule vibrationally excited by an infrared photon, will lose (i.e. dissipate) that energy to the rest of the system by the same mechanism by which normal vibrational energies are exchanged (i.e. fluctuate) between molecules at equilibrium. This insight is the basis of a theoretical description of solution spectroscopy.