Editing Simulated annealing (section)

===Acceptance probabilities===
The probability of making the [[state transition|transition]] from the current state <math>s</math> to a candidate new state <math>s_\mathrm{new}</math> is specified by an ''acceptance probability function'' <math>P(e, e_\mathrm{new}, T)</math>, that depends on the energies <math>e = E(s)</math> and <math>e_\mathrm{new}= E(s_\mathrm{new})</math> of the two states, and on a global time-varying parameter <math>T</math> called the ''temperature''.  States with a smaller energy are better than those with a greater energy.  The probability function <math>P</math> must be positive even when <math>e_\mathrm{new}</math> is greater than <math>e</math>.  This feature prevents the method from becoming stuck at a local minimum that is worse than the global one.

When <math>T</math> tends to zero, the probability <math>P(e, e_\mathrm{new}, T)</math> must tend to zero if <math>e_\mathrm{new} > e</math> and to a positive value otherwise. For sufficiently small values of <math>T</math>, the system will then increasingly favor moves that go "downhill" (i.e., to lower energy values), and avoid those that go "uphill."  With <math>T=0</math> the procedure reduces to the [[greedy algorithm]], which makes only the downhill transitions.

In the original description of simulated annealing, the probability <math>P(e, e_\mathrm{new}, T)</math> was equal to 1 when <math>e_\mathrm{new} < e</math>—i.e., the procedure always moved downhill when it found a way to do so, irrespective of the temperature.  Many descriptions and implementations of simulated annealing still take this condition as part of the method's definition.  However, this condition is not essential for the method to work.

The <math>P</math> function is usually chosen so that the probability of accepting a move decreases when the difference
<math>e_\mathrm{new}-e</math> increases—that is, small uphill moves are more likely than large ones.  However, this requirement is not strictly necessary, provided that the above requirements are met.

Given these properties, the temperature <math>T</math> plays a crucial role in controlling the evolution of the state <math>s</math> of the system with regard to its sensitivity to the variations of system energies. To be precise, for a large <math>T</math>, the evolution of <math>s</math> is sensitive to coarser energy variations, while it is sensitive to finer energy variations when <math>T</math> is small.