Large (and Small) Energy Fluctuations in a Single Classical Degree of Freedom and the Second Law of Thermodynamics

Energy fluctuations in a single classical degree of freedom above the ground state at thermodynamic equilibrium at temperature T are typically of average magnitude ∼kBT. However, we show that the average magnitude of such fluctuations can be much larger (or much smaller) than kBT, indeed, that at least in principle it can be infinite (or arbitrarily close to 0). Nevertheless, the average energy fluctuation magnitude being untypically large (or untypically small) does not violate the second law of thermodynamics. For, if the average magnitude of energy fluctuations is much larger than kBT, then particle motion along the degree of freedom must manifest extreme spatial delocalization. The cost of locating the fluctuating particle along its degree of freedom equals or exceeds the large energy gain obtained upon finding it with an energy of much more than kBT above its ground state. The particle loses as much or more ability to do work via its spatial delocalization than it gains via the energy fluctuation. Similarly, if the average magnitude of energy fluctuations is much smaller than kBT, then the small energy yield obtainable upon locating the particle is compensated for by the small cost of locating it.