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Active matter is composed of large numbers of active "agents", each of which consumes energy in order to move or to exert mechanical forces. Due to the energy consumption, these systems are intrinsically out of thermal equilibrium.

This is the Wikipedia's definition of active matter.

I would like to ask the question that, How are these systems different from a canonical ensemble? In canonical ensemble, sytem exchanges energy with a heat bath. And we treat system plus surroundings as a microcanonical ensemble and derive all quantities of interest. I'm not getting the exact motivation behind this formulation. Any help is appreciated. Thank you

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  • $\begingroup$ In a statistical ensemble the members are not real systems. They are theoretical entities, each of which might be the state of a real system. An ensemble is not a collection of systems. In particular, in an active system, as I understand it, the agents interact. The notion of members of an ensemble interacting has no meaning. $\endgroup$
    – garyp
    Apr 7, 2017 at 19:10
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    $\begingroup$ The issue is not that they interact, in an ideal gas the particles do interact (they collide and exchange energy). In an active system the individual agents require an energy source to behave and interact in the way they do $\endgroup$
    – user126422
    Apr 7, 2017 at 19:52
  • $\begingroup$ @Hugh Mungus Yes you are correct, but it doesn't answer the doubt. $\endgroup$
    – Joe
    Apr 8, 2017 at 9:35
  • $\begingroup$ I do not believe you can write a hamiltonian, because there are hidden variables inside each agent that regulate their behaviour. The ergodicity assumption would also be violated $\endgroup$
    – user126422
    Apr 8, 2017 at 17:15

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The point you raise is essentially why thermodynamic equilibrium is subtle. The primary feature of thermodynamics at equilibrium is detailed balance, which is essentially a consequence of microscopic time reversibility of the dynamics (be it classical or quantum). An active matter system is any system in which detailed balance is broken locally, and hence by definition is out of equilibrium.

To break it down even further, let's look at the canonical ensemble that you point out. At equilibrium, a system in the canonical ensemble exchanges energy with a constant temperature heat bath, providing a standard setup for textbook statistical mechanics. It is imperative for the system to be able to gain and lose energy to the heat bath. This after all is once again a consequence of detailed balance and permits equilibriation of the system to the temperature of the bath. This phrased differently is also a version of the fluctuation dissipation theorem, which arises as the source of fluctuations in an equilibrium system is the same heat bath into which the system dissipates its energy.

Break this, in however small a fashion, and you are out of equilibrium (note that the question of whether the consequences of this broken detailed balance survive on macroscopic scales or an equilibrium like description may be used to study the system, is entirely distinct altogether and can be very system dependent). Suppose the individual units gain energy from a reservoir, but dissipate their energy to a different heat bath. The energy source and sink being distinct (say at different temperatures), we maintain a persistent energy flux through the unit, breaking detailed balance and pushing you out of equilibrium. This is after all what would happen if you placed a gas in a box, say, in contact with two thermal baths at opposite ends of the box. When the bath temperatures differ, a heat current flows from the hot to the cold end and we obtain a rather canonical non-equilibrium steady state. If you had the same setup in the presence of gravity (acting down), with the cold end above the hot one, this leads to Rayleigh-Benard convection, a classic example of a non-equilibrium pattern forming system (see Cross, Hohenberg, Rev. Mod. Phys. 65, 851 (1993), for example).

The additional fuss in an active system is that the individual units consume energy to move persistently or exert forces on the surrounding medium. In order to do so, each unit extracts work from the energy reservoir, instead of the energy flux simply being heat (I am shamelessly glossing over many details as the definitions of both heat and work are tricky, but this is gross picture to have in mind). So, active matter systems are a special class of non-equilibrium systems where the breaking of detailed balance is local by doing work to maintain local persistent motion. Most previous studies of non-equilibrium systems focussed on systems driven at the boundary by an external field or a thermal gradient (by being in contact with two heat baths say), instead, in an active matter system, the drive is local. This turns out to make a big difference leading to spectacular phenomena impossible at equilibrium, like phase separation in the absence of attraction (Phys. Rev. Lett. 108, 235702, 2012; Phys. Rev. Lett. 100, 218103, 2008), long-ranged polar order in 2d evading the Mermin-Wagner theorem (Phys. Rev. Lett. 75, 4326, 1995; Phys. Rev. Lett. 75, 1226, 1995), super-poissonian number statistics in a bulk phase with finite compressibility (Science 317.5834 (2007): 105-108; EPL (Europhysics Letters) 62.2 (2003): 196) and the absence of pressure being a state variable (Nature Physics 11.8 (2015): 673-678). There are many more fascinating results as is obvious from the rapidly increasing interest and work done in this field in the past two decades or so.

At a basic level, one might argue that active systems are just like any other (name your favourite) non-equilibrium system. The hope though is that, instead of going behind a generic description of all non-equilibrium systems, which is most surely hopeless, by restricting ourselves to a this relatively special class, we might gain insight into some of the organizational principles of matter without detailed balance. The other attractive feature is the relevance to biology as living matter often also moves, so hopefully this framework might be useful in understanding the physics of biological organization as well.

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    $\begingroup$ I have read a couple of models of active systems, usually involving Langevin equations, for example to model the cytoplasm, or swarms of bacteria. But I have not found yet a single derivation of the equations of an active system in these contexts, starting from clear and well founded physical assumptions. Rather, they just assume the form of the Langevin equation from the start, or borrow from popular models used in other fields. For me, it is not obvious why the particles in these systems should receive a random force push, instead of a random energy unit. $\endgroup$
    – a06e
    Sep 30, 2017 at 23:28
  • $\begingroup$ Perhaps you can point out some references where the dynamical equations are derived, particularly in some context that resembles the cell, or swarms of cells? If you know about this, I'd appreciate it. $\endgroup$
    – a06e
    Sep 30, 2017 at 23:28
  • $\begingroup$ @Becko I'm not sure your question has an answer. There is some theory connecting Langevin equations to eliminated degrees of freedom (see Kubo's "Statistical Mechanics II or the classic papers by Mori), but I have not seen this used in context of active matter. You may look in the appendix here arxiv.org/pdf/cond-mat/0310384.pdf for a more "from-scratch" derivation of an active matter model $\endgroup$ Feb 18, 2021 at 17:38

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