The previous post mentioned some work on minimal mathematical models for combustion engines, and ended with a link to our first published paper on this subject. That's the paper in which we introduce the class of dynamical systems that we call

The

Our

*Hamiltonian daemons.*The*Hamiltonian*part of this name refers to William Rowan Hamilton's formulation of classical mechanics. Hamilton's restatement of all Newton's**F**= m**a**stuff dates from 1833 but is still the dominant dialect in physics today, mainly because it adapts well to quantum mechanics.The

*daemon*part of our name for our systems is partly meant to recall Maxwell's Demon. The devout Presbyterian Maxwell wasn't thinking of an evil spirit: he meant the kind of benevolent natural entity that was the original meaning of the Latin word*daemon.*Maxwell's daemon was a (purely hypothetical) tiny being who could manipulate individual gas atoms as they flew by him. Physicists still think about Maxwell's daemon today, as a way to express questions about the microscopic limits of thermodynamics.Our

*daemon*name is also partly in analogy to Unix daemons, which are little processes that operate autonomously to manage routine tasks without user input. Hamiltonian daemons are mathematical representations of small, closed physical systems which, without any external power or control, can exhibit steady transfer of energy from fast to slow degrees of freedom. In this way they are ultimate microscopic analogs of combustion engines, and learning about them may teach us about the role of thermodynamics on microscopic scales.A simple Hamiltonian daemon: like a tiny car driving uphill. |

To illustrate how daemons work we took a simple model that resembles a tiny car that tries to drive up an infinite hill. The car's engine has to start in the old-fashioned way of getting it turning by pushing the car, so the car starts out already rolling uphill at some speed. Since it's going uphill, it slows down; some of the time (the dashed curve), the engine fails to ignite and the car just keeps slowing down until it rolls back down the hill.

Other times, however, the engine catches (solid curve). The car then keeps on driving uphill at a steady speed, expending high-frequency fuel. Since our simple hill is infinite, when the fuel runs out the car will eventually roll back down, but if we add the complication of making the hill level off at some point, the car can climb to the top and then drive on to some goal.

Other times, however, the engine catches (solid curve). The car then keeps on driving uphill at a steady speed, expending high-frequency fuel. Since our simple hill is infinite, when the fuel runs out the car will eventually roll back down, but if we add the complication of making the hill level off at some point, the car can climb to the top and then drive on to some goal.

SchrÃ¶dinger's Cat as a power source? |

It turns out that the quantum daemon behaves rather like a random daemon. Sometimes the engine ignites, and sometimes it doesn't—even if you do

*everything exactly the same*every time. The quantum daemon also burns its fuel quantum by quantum, and it keeps driving uphill by giving itself a series of kicks that suddenly bump up its speed. After every such kick, there is a chance that the quantum engine will spontaneously stall, and refuse to run further, even if there is fuel left. So over time the quantum probability distribution for the height of the car develops a series of branches, as in the figure here.The quantum daemon burns fuel in quantized steps, and can randomly stall. |

It's not just randomness going on at every kick, though. If you look closely you can see interference fringes in the probability distribution. Since the whole daemon system is closed, the evolution is actually all unitary. Every branch of the figure, which represents a different possible trajectory of the car, is in coherent quantum superposition. Since there is more fuel left if the engine stalls sooner, but less fuel if the engine goes on longer, the superposition of all the branches is a total quantum state with high quantum entanglement between fast and slow degrees of freedom. Since the number of branches grows over time as the quantum daemon operates, the von Neumann entropy of the slow degrees of freedom increases.

Readers who are familiar with quantum mechanics may want to read more in Physical Review E

Readers who are familiar with quantum mechanics may want to read more in Physical Review E

**96**, 012119 (2017), or in the free and legal ArXiv e-print version. Our first paper on daemons (Phys. Rev. E**94**, 042127 (2016)) is also on ArXiv.