Temperature in an isolated system is defined as: $$\frac{1}{T} = -\frac{\partial{S(E,V,N)}}{\partial{E}} $$ But I wonder how one can generalize this to a random system. Or for instance to a point in a system. Because in these books about statistical physics they talk often about "temperature gradients in a system", but for these to exist, temperature has first to be defined in every point (although I can't find a general definition).
I hope someone can help me out.
Answer
Length scales are not accounted for properly in your question. When you have a system at local equilibrium where a temperature gradient can be defined then each "point" in this description contains say $10^{10}$ molecules and can be seen as a thermostatistical system at equilibrium. We call that "local" equilibrium because intensive quantities such as temperature and chemical potential might not be uniform throughout the whole system i.e. they may vary from one "point" to another. There are evolution equations of these mesoscopic quantities that deal with such local equilibrium situations. The simplest are the Fourier (for temperature) and the Fick (for particle density) equations but they can be derived from more general equations with a collision kernel such as e.g. the Boltzmann equation.
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