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I've encountered this notion many times that turbulence is non-equilibrium (e.g., Falkovich & Sreenivasan, Physics Today, 59, 2006).

However, I am confused about what does it mean by "non-equilibrium"? Non-equilibrium in terms of what? In the sense of thermal equilibrium/non-equilibrium? In Sreenivasan (1999, Rev. Mod. Phys., 71), he says that:

"... turbulence is a paradigm in non-equilibrium statistical physics, in which fluctuations and macroscopic space-time structure coexist."

but this is somewhat a vague explanation (to me at least). So could anyone explain a little bit more of the meaning of non-equilibrium in this context?

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I believe that indeed equilibrium/nonequilibrium here refers to thermal equilibrium/nonequilibrium. Informally, a system is in equilibrium if you leave it be and wait long enough, so that all currents (of energy, particles, etc) flowing through the system quiet down. Take a glass of water and wait, and it equilibrates with its surrounding. However, if you put the water in a pot and heat from a stove below while the top is being cooled by air, then even if the water reaches a steady state where you cannot see any macroscopic motion, energy is flowing from the bottom to the top, and this steady state is a nonequilibrium one.

Turbulent states occur when you keep pumping energy in and it is continually being dissipated, so energy flows through the system (for example, you stir the fluid on a macroscpic scale, and energy "cascades" down to smaller scale eddies until at very small eddies it is being dissipated).

More precisely, the common definition of an equilibrium system is one that is statistically time-reversal symmetric: you cannot tell by looking at a video of the dynamics whether the video is going forward or backward. If you model your system as a Markov process (e.g., by using a Langevin equation), then equilibrium means that the process is "reversible", or satisfies the detailed balance condition. Turbulent fluids do not satisfy these conditions precisely because there is an energy flow.

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    $\begingroup$ Thanks for the reply, it's really helpful. The energy cascade happens from small wavenumbers (large scale) to large wavenumbers (small scale). So I am wondering: if a system (it can be any system) has energy flux from one mode to another, or the energy distribution is not equipartition, can this system be characterized as nonequilibrium? $\endgroup$
    – TurbPhys
    May 25 '20 at 2:39

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