Entropic force

In physics, an entropic force acting in a system is a force resulting from the entire system's thermodynamical tendency to increase its entropy, rather than from a particular underlying microscopic force.^[1]

For instance, the internal energy of an ideal gas depends only on its temperature, and not on the volume of its containing box, so it is not an energy effect that tends to increase the volume of the box as gas pressure does. This implies that the pressure of an ideal gas has an entropic origin. ^[2]

What is the origin of such an entropic force? The most general answer is that the effect of thermal fluctuations tends to bring a thermodynamic system toward a macroscopic state that corresponds to a maximum in the number of microscopic states (or micro-states) that are compatible with this macroscopic state. In other words, thermal fluctuations tend to bring a system toward its macroscopic state of maximum entropy. ^[2]

Mathematical formulation

In the canonical ensemble, the entropic force $\mathbf F$ associated to a macrostate partition $\{\mathbf{X}\}$ is given by:^[3][4]

$\mathbf{F}(\mathbf{X_0}) = T \nabla_{\mathbf X} S(\mathbf{X})|_{\mathbf X_0}$

where $T$ is the temperature, $S(\mathbf{X})$ is the entropy associated to the macrostate $\mathbf{X}$ and $\mathbf{X_0}$ is the present macrostate.

Examples

Brownian motion

The entropic approach to Brownian movement was initially proposed by R. M. Neumann,^[3][5] Neumann derived the entropic force for a particle undergoing three-dimensional Brownian motion using the Boltzmann equation, denoting this force as a diffusional driving force or radial force. In the paper, three example systems are shown to exhibit such a force:

• electrostatic system of molten salt
• surface tension and
• Elasticity of rubber.

Polymers

A standard example of an entropic force is the elasticity of a freely-jointed polymer molecule.^[5] For an ideal chain, maximizing its entropy means reducing the distance between its two free ends. Consequently, a force that tends to collapse the chain is exerted by the ideal chain between its two free ends. This entropic force is proportional to the distance between the two ends. ^[2][6]

Hydrophobic force

Water drops on the surface of grass.

Another example of an entropic force is the hydrophobic force. At room temperature, it partly originates from the loss of entropy by the 3D network of water molecules when they interact with molecules of dissolved substance. Each water molecule is capable of

• donating two hydrogen bonds through the two protons
• accepting two more hydrogen bonds through the two sp³-hybridized lone pairs

Therefore, water molecules can form an extended three-dimensional network. Introduction of a non-hydrogen-bonding surface disrupts this network. The water molecules rearrange themselves around the surface, so as to minimize the number of disrupted hydrogen bonds. This is in contrast to hydrogen fluoride (which can accept 3 but donate only 1) or ammonia (which can donate 3 but accept only 1), which mainly form linear chains.

If the introduced surface had an ionic or polar nature, there would be water molecules standing upright on 1 (along the axis of an orbital for ionic bond) or 2 (along a resultant polarity axis) of the four sp³ orbitals.^[7] These orientations allow easy movement, i.e. degrees of freedom, and thus lowers entropy minimally. But a non-hydrogen-bonding surface with a moderate curvature forces the water molecule to sit tight on the surface, spreading 3 hydrogen bonds tangential to the surface, which then become locked in a clathrate-like basket shape. Water molecules involved in this clathrate-like basket around the non-hydrogen-bonding surface are constrained in their orientation. Thus, any event that would minimize such a surface is entropically favored. For example, when two such hydrophobic particles come very close, the clathrate-like baskets surrounding them merge. This releases some of the water molecules into the bulk of the water, leading to an increase in entropy.

Another related and counter-intuitive example of entropic force is protein folding, which is a spontaneous process and where hydrophobic effect also plays a role.^[8]

Directional entropic force

Entropic forces also occur in the physics of gases and solutions, where they generate the pressure of an ideal gas (the energy of which depends only on its temperature, not its volume), the osmotic pressure of a dilute solution, and in colloidal suspensions, where they are responsible for the crystallization of hard spheres.

In nano and colloidal science, entropic forces usually come from the effect of depletion, where small particles induce crystallization of bigger ones.

Even in the absence of depletion, however, scientist Sharon Glotzer and collaborators recently conjectured that directional entropic forces could be responsible for the alignment of facets observed prior to the assembly and/or crystallization of systems of polyhedral nano and colloidal particles.^[9] This was later proven to be correct^[10][11] and allowed for the development of a roadmap for the assembly of polyhedral particles into atomic isostructures.^[12]

Controversial examples

Some forces that are generally regarded as conventional forces have been argued to be actually entropic in nature. These theories remain controversial and are the subject of ongoing work. Matt Visser, professor of mathematics at Victoria University of Wellington, NZ in "Conservative Entropic Forces" ^[13] criticizes selected approaches but generally concludes:

There is no reasonable doubt concerning the physical reality of entropic forces, and no reasonable doubt that classical (and semi-classical) general relativity is closely related to thermodynamics. Based on the work of Jacobson, Thanu Padmanabhan, and others, there are also good reasons to suspect a thermodynamic interpretation of the fully relativistic Einstein equations might be possible.

Gravity

In 2009, Erik Verlinde argued that gravity can be explained as an entropic force.^[14] It claimed (similar to Jacobson's result) that gravity is a consequence of the "information associated with the positions of material bodies". This model combines the thermodynamic approach to gravity with Gerard 't Hooft's holographic principle. It implies that gravity is not a fundamental interaction, but an emergent phenomenon.^[15]

Other forces

In the wake of the discussion started by Verlinde, entropic explanations for other fundamental forces have been suggested,^[13] including Coulomb's law,^[16][17][18] the electroweak and strong forces.^[19] The same approach was argued to explain dark matter, dark energy and Pioneer effect.^[20]

Links to adaptive behavior

It was argued that causal entropic forces lead to spontaneous emergence of tool use and social cooperation.^[21][22][23] Causal entropic forces by definition maximize entropy production between the present and future time horizon, rather than just greedily maximizing instantaneous entropy production like typical entropic forces.

A formal simultaneous connection between the mathematical structure of the discovered laws of nature, intelligence and the entropy-like measures of complexity was previously noted in 2000 by Andrei Soklakov^[24] in the context of Occam's razor principle.

See also

• Colloids
• Nanomechanics
• Data clustering
• Depletion force
• Entropic elasticity of an ideal chain

References

1. A history of thermodynamics: the doctrine of energy and entropy by Ingo Müller, p115
2. Taylor; Tabachnik (2013). "Entropic forces—making the connection between mechanics and thermodynamics in an exactly soluble model". European Journal of Physics 34 (3).
3. Neumann RM (1980). "Entropic approach to Brownian movement". American Journal of Physics 48 (5): 354. Bibcode:1980AmJPh..48..354N. doi:10.1119/1.12095.
4. On the origin of gravity and the laws of Newton, Erik Verlinde
5. Neumann RM (1977). "The entropy of a single Gaussian macromolecule in a noninteracting solvent". The Journal of Chemical Physics 66 (2): 870. Bibcode:1977JChPh..66..870N. doi:10.1063/1.433923.
6. Smith, SB; Finzi, L; Bustamante, C (1992). "Direct mechanical measurements of the elasticity of single DNA molecules by using magnetic beads". Science 258 (5085): 1122–6. Bibcode:1992Sci...258.1122S. doi:10.1126/science.1439819. PMID 1439819.
7. Encyclopedia of Life Science Article on Hydrophobic Effect; See Figure 4: http://xibalba.lcg.unam.mx/~rgalindo/bioquimica/BQPosgrado2011/I%20FQ%20repaso/HydrophobicEffect.pdf
8. http://www.wiley.com/college/pratt/0471393878/student/review/thermodynamics/7_relationship.html
9. "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces" (PDF). ACS. Archived from the original on 2011-12-01. Retrieved 23 June 2012.
10. "Unified Theoretical Framework for Shape Entropy in Colloids" (PDF). Archived from the original (PDF) on 2013-09-03. Retrieved 20 October 2013.
11. "A Directional Entropic Force Approach to Assemble Anisotropic Nanoparticles into Superlattices" (PDF). Archived from the original (PDF) on 2013-09-03. Retrieved 13 Jan 2014.
12. "Structural Diversity and the Role of Particle Shape and Dense Fluid Behavior in Assemblies of Hard Polyhedra" (PDF). Archived from the original (PDF) on 2012-02-10. Retrieved 23 June 2012.
13. Visser, Matt. "Conservative entropic forces". arXiv:1108.5240.
14. E.P. Verlinde. "On the Origin of Gravity and the Laws of Newton". JHEP 04, 29 (2011). arXiv:1001.0785. Bibcode:2011JHEP...04..029V. doi:10.1007/JHEP04(2011)029.
15. E.P. Verlinde. "On the Origin of Gravity and the Laws of Newton". JHEP 04, 29 (2011). arXiv:1001.0785. Bibcode:2011JHEP...04..029V. doi:10.1007/JHEP04(2011)029.
16. http://arxiv.org//abs/1001.4965, Coulomb Force as an Entropic Force, T. Wang
17. http://arxiv.org//abs/0809.4631, Simple field theoretical approach of Coulomb systems. Entropic effects, D. di Caprio, J.P. Badiali, M. Holovko
18. http://arxiv.org//abs/1009.5561, Entropic Corrections to Coulomb's Law, A. Sheykhi, S. H. Hendi
19. http://arxiv.org//abs/1008.4147, Emergent Gauge Fields, Peter G.O. Freund
20. http://arxiv.org//abs/1009.1506 Unification of Dark Matter and Dark Energy in a Modified Entropic Force Model, Zhe Chang, Ming-Hua Li, Xin Li
21. Wissner-Gross, A.D.; Freer, C.E. (2013). "Causal Entropic Forces" (PDF). Physical Review Letters 110 (16). doi:10.1103/PhysRevLett.110.168702.
22. http://arxiv.org/abs/1308.4375, Comment on Phys. Rev. Lett. 110, 168702 (2013): Causal Entropic Forces, E. Canessa
23. http://arxiv.org/abs/1312.4185, Comment: Causal entropic forces, H.J. Kappen
24. Andrei N. Soklakov, "Occam's razor as a formal basis for a physical theory" (arXiv:math-ph/0009007, September 2000; Foundations of Physics Letters, 2002), "Complexity analysis for algorithmically simple strings" (arXiv:cs/0009001, September 2000).