**1. Introduction: Entropy Force**

About thirty years ago, Prigogine [1] attempted a microscopic theory of entropy assuming that, by some quantum process, seeds of entropy could be generated. Such a hypothetical process would, in the early universe, possibly lay down the direction of time. Unfortunately, so far, such microscopic sources of entropy have not been confirmed. It seems that they can hardly be expected because quantum uncertainty itself is a stochastic process, which by its own nature does not contain any direction. It is hard to believe that it could lead to entropy production if not aided by some kind of dissipative interaction. Entropy is a thermodynamic concept, which by itself requires an underlying dynamics, which allows for the presence of many states that a system consisting of many subsystems, components, particles would be able to occupy.

More recently, it has been speculated [2] that that kind of a mesoscopic entropy in quantum string theory could cause gravity to emerge from the action of a quantum entropic force as a gradient of entropy generated in string interactions, intended to provide a physical basis for the so-called modified Newtonian gravity, which proposes that Newton's law should be corrected on the large scales to eliminate the problem of dark matter in astronomy.

From a completely different point of view, the idea of an entropy force has been picked up in the discussion of maximum entropy methods in prediction theory [3,4] in open systems where the probabilistic version of entropy depends on space and time, propagates into the future and, thus, has a finite gradient in space and time, which is interpreted as force. It apparently is capable of allowing, based on maximization of entropy, predicting the time evolution of the system, an interesting and possibly far-reaching predictive concept. In a sufficiently small closed system, it necessarily must describe the evolution of entropy towards a finite thermal state of maximum entropy. Recently, entropy forces have also been applied to molecular dynamics in proteins ([5] and the references therein). Spatial smallness is required by causality to enable synchronization. Therefore, the concept applies to the universe on the cosmological timescale only in order to allow for homogenization of entropy. On any local scale, the entropy produced in the classical system represents a localized excess in entropy. If not artificially confined, this excess tends to expand and affect its environment. This necessarily generates a local entropy force, a classical force that should not be mixed up with the above-mentioned entropic force in string systems. This force follows from the first law in thermodynamics:

$$dE = TdS - PdV\tag{1}$$

which relates the three different forms of energy *E*, pressure *PV*, and entropy *TS*. Gradients in energy, pressure, and temperature are known to be forces, and the gradient of volume causes dispersion, flows, and forces, for instance in charged systems.

In a similar vein, a gradient in entropy corresponds to a collective macroscopic effect as the entropy tends to expand and maximize. This is a purely macroscopic effect indeed because, similar to density/volume and pressure, the entropy *S* is defined only for macroscopic systems, consisting of a large number of subsystems, to which finite temperature and density can be assigned and which occupy a finite volume. In the first law, it is only the energy *E* that maintains its meaning also in the microscopic world down to only one particle, to which assigning temperature makes no sense. The entropy potential *U* = *TS* indeed is not just a thermodynamic potential; it is also a real potential always being positive and thus repulsive. The entropy force is then given as its gradient:

$$\mathbf{F} = -\nabla \mathbf{U} \tag{2}$$

as usually taken negative. It consist of two parts, a thermal force −*S*∇*T*, which is of no interest here, and the genuine entropy force:

$$\mathbf{F}\_{\mathcal{S}} = -T\nabla \mathcal{S} \tag{3}$$

This might look trivial; however, it is not, as we will demonstrate below with a particular example: the black hole.

However, before proceeding, we recall that, since both *T* and *S* are positive definite, the entropy force is repulsive in the direction negative to the gradient of entropy. This means that an accumulation of entropy at some location, if not artificially confined to a box, will act outward. Adopting an interpretation of entropy as disorder, which by no means is generally justified, thus implies that disorder tends to infect its external region. It has the tendency to expand.

With temperature *T* in energy units, the entropy *S* has no dimension. Moreover, the product of temperature and entropy is a scalar function with the dimension of a potential. For scalar temperature, i.e., at temperature isotropy, *S* is also a scalar. Under conditions of anisotropic temperature, the inverse temperature becomes a vector [6], and thus, *S* becomes a vector as well (more generally, both become tensors). In the interest of simplicity, we do not consider this case in the following.

This entropy force does not depend on particle mass or charge, at least not explicitly. Mass is contained in temperature and energy, but there is no explicit reference to it in the definition of the entropy force. Thus, for a given temperature, all particles independent of their properties will be subject to the same entropy force. In this sense, the entropy force is a general mechanical force seeking to restore smoothness in disorder on a higher level of disorder, completely independent of which kind of particles have contributed to the inhomogeneity in disorder.

By its nature, the above entropy force is a long-range force. It does not compete with the Coulomb force on the short scales. From the first law, one realizes that the main force it competes with is the pressure gradient. Both are proportional to the temperature, which thus drops out when comparing the forces. Both are proportional to the density gradient. Thus, in the presence of changing volume, the entropy force adds to or diminishes the effect of the pressure force.
