Relaxation equilibrium, temperature-time dependence

At high temperatures, a nanoparticle is in a superparamagnetic state with thermal equilibrium properties as described in the previous section. At low temperatures, the magnetic moment is blocked in one potential well with a small probability to overcome the energy barrier, while at intermediate temperatures, where the relaxation time of a spin is comparable to the observation time, dynamical properties can be observed, including magnetic relaxation and a frequency-dependent ac susceptibility. 

A superstatistical equilibrium distribution is written as a superposition of Boltzmann distributions with different temperatures. We showed that the excess heat could be written as a superposition of correlation functions with different temperatures using the generalized fluctuation-dissipation theorem. When a relaxation time depends on a temperature, we can expect various behaviors for the area of the hysteresis loop from the fluctuation-dissipation theorem. 

With t-jump, the temperature of a sample is rapidly changed. Thus, any temperature-dependent equilibrium is perturbed and the concentrations of reactants and products must be altered to the values necessary for equilibrium at the new temperature. If the temperature change is more rapid than the system can react, then the relaxation of the concentration alterations can be measured. This time dependence, which is usually exponential, can then be utilized to derive rate constants at the final temperature for the involved chemical reactions (Turner, 1986). 

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