Also, due to Rudolf Clausius, is the simplest formulation of the second law, the heat formulation or Clausius statement:

Heat generally cannot spontaneously flow from a material at lower temperature to a material at higher temperature.

Informally, "Heat doesn't flow from cold to hot (without work input)", which is obviously true from everyday experience. For example in a refrigerator, heat flows from cold to hot, but only when aided by an external agent (i.e. the compressor). Note that from the mathematical definition of entropy, a process in which heat flows from cold to hot has decreasing entropy. This can happen in a non-isolated system if entropy is created elsewhere, such that the total entropy is constant or increasing, as required by the second law. For example, the electrical energy going into a refrigerator is converted to heat and goes out the back, representing a net increase in entropy.

The exception to this is in statistically unlikely events where hot particles will "steal" the energy of cold particles enough that the cold side gets colder and the hot side gets hotter, for an instant. Such events have been observed at a small enough scale where the likelihood of such a thing happening is large enough.^[2] The mathematics involved in such an event are described by fluctuation theorem.