# Second law of thermodynamics in linear response theory

I am wondering about the appearance of irreversibility in the response functions or equivalently the correlation functions in a statistical mechanics system. The main principle that I have seen where this is introduced is the principle that the power absorbed by external forces must be positive (see for instance in Chaikin and Lubensky 7.6.3). I am wondering if someone can draw a connection to more familiar ideas such as the second law of thermodynamics in terms of entropy, or otherwise convince me that this principle must be valid.

The idea behind this principle is we are treating external forces $h(x,t)$ as some arbitrary classical field that appears as a time dependent term in the Hamiltonian coupled to some field $\phi(x)$ which is an observable of the system.

Now we take the system to be in a state which is the equilibrium state for $t=0$ but evolves according to the full Hamiltonian with the term involving $h$ taken into account. Then we look at the average power absorbed, which is time derivative of the expected energy $E$ of the system averaged over the time scale of the perturbation. The principle is that $$\overline{\frac{dE}{dt}}\geq0.$$ This statement lets us derive some conditions on the response functions, which are in turn equivalent to conditions on correlation functions through the fluctuation-dissipation theorem.

I am wondering how exactly this principle which introduces dissipation is justified.

• If the theory is linear you can do this for every possible spectrum frequency by frequency, so you end up integrating positive values. In linear systems the individual frequencies do not interact (which, OTOH, denies you access to the ergodic hypothesis). Am I missing something? Shouldn't the average power be zero in thermal equilibrium? – CuriousOne Mar 27 '16 at 0:49
• The example I've seen given is that of a harmonic oscillator. If there is a dissipative term and a driving oscillation, in a steady state the power transfered to the oscillator (and then dissipated) will be positive. It seems what they are doing is flipping the argument around and starting with this positive power principle and then deriving the sign of dissipation from that. – octonion Mar 27 '16 at 3:11
• I think all of that is pretty much covered by the fluctuation-dissipation theorem, isn't it? – CuriousOne Mar 27 '16 at 4:36
• No the fluctuation-dissipation theorem is connecting correlation functions in equilibrium to the response function (which describes how expectation values change in the presence of this external field $h$). The context of my question is an argument leading to certain conditions on the response function which are characteristic of "dissipation," but I think they are independent of the fluctuation-dissipation theorem itself. My question is how this condition on power can be related to more familiar conditions involving entropy. – octonion Mar 27 '16 at 7:34
• Of course if there is an argument using the fluctuation-dissipation theorem that describes the appearance irreversibility in terms of correlation functions, that would interest me too. – octonion Mar 27 '16 at 7:40