Entropy force

The entropy force or entropic force has its cause in the thermal movement of the particles under an externally applied force.

Mathematical formulation

The entropic force assigned to the macrostate is given by the negative gradient of the contribution of the entropy to ${\ displaystyle {\ vec {X}} _ {0}}$ ${\ displaystyle {\ vec {F}} ({\ vec {X}} _ {0})}$ ${\ displaystyle - \ nabla}$ ${\ displaystyle -T \ cdot S}$ ${\ displaystyle S}$

Free energy in the canonical ensemble ${\ displaystyle F = UT \ cdot S}$
Free enthalpy in the isothermal-isobaric ensemble ${\ displaystyle G = HT \ cdot S}$
grand canonical potential in the grand canonical ensemble , ${\ displaystyle \ Omega = UT \ cdot S- \ mu \ cdot N}$

each:

{\ displaystyle {\ begin {aligned} {\ vec {F}} ({\ vec {X}} _ {0}) & = - \ nabla _ {\ vec {X}} \ left (-T \ cdot S ({\ vec {X}}) \ right) | _ {{\ vec {X}} _ {0}} \\ & = T \ cdot \ nabla _ {\ vec {X}} \; S ({\ vec {X}}) | _ {{\ vec {X}} _ {0}} \ end {aligned}}}

With

${\ displaystyle {\ vec {X}} _ {0}}$ the macrostate under consideration
${\ displaystyle T}$ the temperature
${\ displaystyle S ({\ vec {X}})}$ the entropy associated with the macrostate . ${\ displaystyle {\ vec {X}}}$

The entropic force thus acts along the steepest rise in entropy in the space of macrostates. ${\ displaystyle T> 0}$

Examples

Polymers

The tensile force of a rubber or an individual polymer is based on entropy, i.e. indirectly on the energy of the heat bath , in contrast to a hard spring, the force of which is due to the stored distortion energy . With polymers, the constraint is given by the fact that it is stretched due to the external tensile force and thus fewer microscopic configurations are available.

The entropy force can have a different representation depending on the underlying polymer model. What they all have in common is that for moderately stretched polymers, Hooke's law with a temperature-dependent spring constant applies. At the temperature so acts on a polymer having segments of the effective length and the end-to-end distance of the force ${\ displaystyle D (T)}$ ${\ displaystyle T}$ ${\ displaystyle N}$ ${\ displaystyle b}$ ${\ displaystyle x}$

{\ displaystyle F = D (T) \ cdot x = {\ frac {3 \ cdot k _ {\ mathrm {B}} \ cdot T} {N \ cdot b ^ {2}}} \ cdot x}

with the Boltzmann constant ${\ displaystyle k _ {\ mathrm {B}}.}$

See also: Entropy elasticity

osmosis

Entropic forces also occur in osmosis . The driving force behind osmotic processes is a gradient in the chemical potential between the phases separated by a semipermeable membrane . The semipermeable membrane creates compulsion: here too, the configuration space for particles that cannot cross the membrane is restricted. As the solvent flows into the phase with lower potential, the entropy of the overall system increases by the corresponding entropy of mixing .

Gravity

Erik Verlinde described gravity as an entropic force in 2009. In string theory , space-time coordinates are fields that arise from an underlying theory. In analogy to the description of black holes , the holographic principle is applied to mass-containing space-time domains and Newton's gravitational equation is derived. Due to its behavior in areas of extremely weak gravity ( interstellar space ), entropic gravity offers a possible explanation for the rotation curves of galaxies without having to resort to dark matter .

Hydrophobic effect

The explanation of the hydrophobic effect , which relies on entropy as an explanation, uses an entropic force as an explanation.

Impoverishment forces

Overlap of the excluded volumes of two large spheres

A depletion force in a bidisperse suspension of e.g. B. hard balls is an entropic force. Forces of impoverishment can be described by the Asakura-Oosawa model (AO model) (within the framework of the approximations of the model). The core of the AO model is that for a system of hard interacting particles in the NVT ensemble, the free energy difference only has the entropic contribution and that this contribution can be obtained using the equation for the entropy of the ideal gas (non-interacting point particles) in the NVT ensemble by taking into account the expansion of the spheres by hand using the idea of excluded volumes. The excluded volumes are the volumes that are not accessible to the centers of gravity of the small particles, since particles with a hard interaction potential cannot overlap in a physical system. If two excluded volumes overlap with the overlap , the small spheres have more freedom of movement: they can now move effectively in the volume . This gives a difference in free energy between a state with an overlapping excluded volume and a state without an overlap: ${\ displaystyle \ Delta V}$ ${\ displaystyle V + \ Delta V}$

{\ displaystyle \ Delta F = -TNk _ {\ mathrm {B}} \ ln \ left ({\ frac {V + \ Delta V} {V}} \ right),}

wherein depends on the geometry of the suspended particles and the considered geometry. For example, for a large sphere in front of a wall (with small spheres as depletion agent), the excluded volume of the large sphere and the wall is given by the volume of the spherical segment of the overlap. ${\ displaystyle \ Delta V}$ ${\ displaystyle \ Delta V}$

Individual evidence

↑ Description of Braun's motion with the aid of entropy; Richard M. Neumann, Am. J. Phys. 48, 354 (1980) doi : 10.1119 / 1.12095
↑ On the origin of gravity and the laws of Newton , Erik Verlinde
↑ Principles of Molecular Recognition, Buckingham, Legon, Roberts, ISBN 0751401250 , pp. 4 and 5, Google Books

[1] Description of Braun's motion with the aid of entropy; Richard M. Neumann, Am. J. Phys. 48, 354 (1980) doi : 10.1119 / 1.12095

[2] On the origin of gravity and the laws of Newton , Erik Verlinde

[3] Principles of Molecular Recognition, Buckingham, Legon, Roberts, ISBN 0751401250 , pp. 4 and 5, Google Books