# Time and Temperature

I found this article very interesting, mostly due to the way a bold prediction – the data stored on this disk will be stable for a million years! – is based completely on probabilistic theory. From the news article:

“The probability that the system will jump in this way is governed by an idea known as Arrhenius law. This relates the probability of jumping the barrier to factors such as its temperature, the Boltzmann constant and how often a jump can be attempted, which is related to the level of atomic vibrations.”

Temperature is a measure of the average thermal energy available to the molecules of the system (kT). Since the particles are moving around and colliding randomly, energy is constantly being exchanged and there is a chance that one will have significantly more energy than the average. In fact, the probability is given by the exponential Boltzmann distribution. If there is an energy barrier (W), that is, an intermediate state higher in energy than the reactants, only the particles that are “thermally activated” can participate. This allows the rate constant for the reaction to be calculated using transition state theory. Basically, the expected time to wait for a reaction to occur scales Exp[W/kT], reflecting the same distribution of energy in the constituent molecules. Since the Boltzmann distribution maximizes the entropy for a given average energy, it is the overwhelmingly the most probable state of affairs. However, there is always (a very small) chance that large enough deviations will occur much faster. So there is no guarantee that you data will be safe a million years hence, but the odds are in your favor if the activation barrier is large enough.

In biology, enzymes control the processes of life by selectively lowing the energy barriers at the appropriate times. Usually this is achieved by stabilizing the transition state (often by having the right charges in the right places). As a result, some reactions that would take millions of years in the absence of enzymes occur in milliseconds. This change by a factor of 10^17 comes from the exponential dependence in the Arrhenius equation.