Sorry if this is something that is well known, not really familiar with modern physics beyond high school / introductory undergrad level. I largely work in deep learning and broadly speaking, you can think of training a deep neural network as an unconstrained optimization (minimization) problem on a loss function whose domain is the space of all parameters that need to be optimized. We now want to reach the global minimum of this loss function and that point in the domain is the model we use (very broad description).

Can I draw analogies of this to physical systems? I can think of all the independent variables of the system as the domain and the net energy of the system as the function we are optimizing over (analogous to the loss function). Since the system will try to go to its state of minimum energy (entropy has to increase and all that), can I figure out the final state of a physical system (and the path it takes), just by looking at its initial state and solving this unconstrained optimization problem? Is this common practice by any means, and if not is there any reason this does not work out or is inefficient?

If this is a valid way of going about it, then is there the problem of stopping at a local minimum? Even though in DL it's rare (statistically saddle points are far more common when you have a large number of independent variables) -- It would make sense that this is a real problem (lower num of independent variables I would imagine), since any change at a local minimum does increase the energy of the system, which is unnatural by thermodynamics. Is this a common issue that actually translates to physical systems?

PS: Apologies if this makes no sense, my idea of physics largely pertains to introductory undergrad physics. If this does make sense though and is actually common practice, would really appreciate links or lectures which tackle this idea.


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In thermodynamics, minimisation is indeed used as a means of finding equilibrium states which are the states the system converges to when left alone. However, what is minimised or maximised depends on the system one is studying. For example, in a closed system, the entropy has a maximum at the equilibrium, as stated by the second law of thermodynamics. If, however, you have a system at fixed temperature, volume and particle number, the free energy will be minimal in the equilibrium.

Along which path in the coordinate space the system will converge to the minimum or maximum is a whole other question. In fact, there is no guarantee that the system will reach that point at all, for example glass and diamond are stable, but quartz and graphite would be the states with the lowest chemical potential and thus thermodynamically preferable. To my knowledge, there is no general recipe (minimisation approach or other) to figure out that path.

This stuff is normally covered in any lecture or book about statistical physics or thermodynamics, sometimes more and sometimes less detailed. There also is a wikipedia article about thermodynamic potentials, that also mentions when which of them is maximal or minimal.

Other parts of physics than thermodynamics can also be formulated as optimisation problems. For example, there is the formulation of Lagrangian mechanics, which uses minimisation of the time-integral over the difference of kinetic and potential energy to determine the trajectory of particles.


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