The second law of thermodynamics has an implicit assumption that all states in an ensemble have roughly the same order of magnitude of probability of receiving energy from another state in the ensemble. Ensembles in thermodynamics have the same mathematical structure as Markov chains, so to put this another way, in order to get the second law of thermodynamics, we have to assume that it is impossible to build an ensemble that behaves like an absorbing Markov chain.
These days however, there are an increasing number of counterexamples to this assumption, like the Graphene Energy Harvesters at the University of Arkansas, and Epicatalytic thermal "diodes" at UC San Diego.
When we don't make this assumption about microcanonical ensembles, then we must use Ramsey Theory to construct our new thermodynamic variables. Existing thermodynamics is still self consistent, but its variables and equations cannot be used to falsify a superset of itself, because those variables cannot apply if the assumptions used to define them don't apply.
These "Maxwell's Zombie" systems have a few things in common: they often involve phase changes, absorbing Markov dynamics, modulation of boundary conditions/varying the boundary of the ensemble, and "biasing potential energy", which is often not used up in the operation of the device, but rather biases the flow of other energy to be harvested. These systems usually involve a process, measurable in terms of electron volts or temperature, which defines the minimum operating temperature of the device as a "virtual" heatsink.
For example, with the Graphene Energy Harvester, an electrostatic charge is placed on the graphene sheet, which vibrates from thermal motion with millivolt energies sufficient to overcome the forward voltages of MIM diodes in a full bridge rectifier. The excess energy harvested from thermal motion becomes trapped in a capacitor, which may be considered the "absorbing state" of the system. But because we know how to use capacitors, we can simply periodically close the switch and use the resulting energy, like water in a hydroelectric reservoir. This process changes the absorbing Markov Chain ensemble from a closed system to an open system.
Another example is the self-charging thermionic capacitor: This time, the biasing energy is the single heat bath the capacitor is immersed in, and the operating temperature is defined in eV by the work functions of the two metal plates involved. If the operating temperature is high enough, the low work function material will emit more electrons than the high work function material, developing a net charge between the plates which may be harvested. Note that unlike a Carnot engine, this device requires no heat sink, the "virtual heat sink" is defined only by the work functions of the two metals. Similar to the Graphene Energy Harvester, the charge on the capacitor may be tapped to do work in an external system.
Most of these devices require extreme operating conditions or semiconductor fabrication capabilities, and so are not accessible to us mere mortals. During the course of this project, I will attempt to construct some devices which are approachable and buildable by the Hackaday community, as well as document the fundamental physics and math involved