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Victoria University of WellingtonTe Whare Wananga o te Upoko o te Ika a Maui

VUWConservative entropic forces

Matt Visser

Gravity as Thermodynamics:Towards the microscopic origin of geometry

ESF Exploratory WorkshopSISSA/ISAS, Trieste, Italy

Monday 5th September 2011

Matt Visser (VUW) Conservative entropic forces gtc2011 1 / 55

Abstract:

VUWEntropic forces mooted as ways to reformulate, retrodict,and perhaps even explain, classical Newtonian gravity.

Newtonian gravity is described by a conservative force,

Implies significant constraints on the entropy and temperature.

Implies real and significant problems for any reasonable variant ofVerlindes entropic gravity proposal.

Though without directly impacting on either Jacobsons orPadmanabhans versions of entropic gravity.

Resolution? Extend the usual notion of entropic force to multipleheat baths with multiple temperatures and multiple entropies?

arXiv: 1108.5240 [hep-th].

VUWMatt Visser (VUW) Conservative entropic forces gtc2011 2 / 55

Outline:

VUW1 Background

2 Conservative entropic forces

3 Verlindes proposal

4 Thermodynamic forces

5 Discussion

VUW

Matt Visser (VUW) Conservative entropic forces gtc2011 3 / 55

Background:

Background

Matt Visser (VUW) Conservative entropic forces gtc2011 4 / 55

Background:

I shall not attempt to derive or justify an entropic interpretationfor Newtonian gravity.

Rather I shall ask the converse question:

Assuming that Newtonian gravity can be described by anentropic force, what does this tell us about the relevanttemperature and entropy functions of the assumedthermodynamic system?

Matt Visser (VUW) Conservative entropic forces gtc2011 5 / 55

Background:

Start from the definition of an entropic force

F = T S .

Demand that this entropic force reproduces the conservative force lawof Newtonian gravity

F = .

This places some rather strong constraints on the functional form ofthe temperature and entropy.

Matt Visser (VUW) Conservative entropic forces gtc2011 6 / 55

Conservative entropic forces:

Conservative entropic forces

Matt Visser (VUW) Conservative entropic forces gtc2011 7 / 55

Conservative entropic forces:

There is no doubt that entropic forces exist.

There are numerous physical examples.

The most well-known are:

elasticity of a freely jointed polymer;hydrophobic forces;osmotic forces;colloidal suspensions;binary hard sphere mixtures;molecular crowding/depletion forces.

Classically reversible.

Matt Visser (VUW) Conservative entropic forces gtc2011 8 / 55

Conservative entropic forces:

Can entropic forces be used to mimic Newtonian gravity?

More generally:Can you mimic any conservative force derivable from a potential?

Can this be done in a manner consistent with Verlindes specificproposal?

For definiteness we shall focus on two specific cases:

1 A single particle interacting with an externally specified potential.(Single position variable r.)

2 A many-body system of n mutually interacting particles.(n position variables ri , for i {1 . . . n}.)

Matt Visser (VUW) Conservative entropic forces gtc2011 9 / 55

Conservative entropic forces: 1-body

Single body interacting with an externally specified potential

F(r) = (r).

Assume this can be mimicked by an entropic force

F(r) = T (r) S(r).

Implies(r) = T (r) S(r).

Without any calculation, since ||S , this implies that the levelsets of the potential are also level sets of the entropy.

Implies the entropy is some function of the potential.

Matt Visser (VUW) Conservative entropic forces gtc2011 10 / 55

Conservative entropic forces: 1-body

Take the curl, we also see T ||S .So level sets of the temperature are also level sets of the entropy.

Introduce some convenient normalization constants E and T,related by E = kB T.

General solution:

T (r) =T

f ((r)/E); S(r) = kB f ((r)/E).

Here f (x) is an arbitrary monotonic function,and f (x) = df /dx is its derivative.Verify solution is correct by using the chain rule.

Monotonicity of f (x) required to avoid a divide-by-zero error.

Matt Visser (VUW) Conservative entropic forces gtc2011 11 / 55

Conservative entropic forces: 1-body

Summary:

T (r) =T

f ((r)/E); S(r) = kB f ((r)/E).

Very simple and very general constraint on the temperature andentropy of any thermodynamic system capable of mimicking anexternally imposed conservative force.

Very powerful constraint.

Very problematic for Verlindes proposal.

Matt Visser (VUW) Conservative entropic forces gtc2011 12 / 55

Conservative entropic forces: n-body

Consider n bodies mutually interacting via a conservative force.

Argument very similar.

Just enough difference to make an explicit exposition worthwhile.

The force on the i th particle is

Fi (r1, . . . , rn) = i (r1, . . . , rn).

Assume this can be mimicked by an entropic force

Fi (r1, . . . , rn) = T (r1, . . . , rn) iS(r1, . . . , rn).

Implies

i (r1, . . . , rn) = T (r1, . . . , rn) iS(r1, . . . , rn).

Matt Visser (VUW) Conservative entropic forces gtc2011 13 / 55

Conservative entropic forces: n-body

Without any calculation, i we have i ||iS .Implies that the level sets of the potential are also level sets of theentropy.

Implies that the entropy is some function of the potential.

Take the curl (with respect to the variable ri ).

i we have iT ||iS .So level sets of the temperature are also level sets of the entropy.

Matt Visser (VUW) Conservative entropic forces gtc2011 14 / 55

Conservative entropic forces: n-body

As in the 1-body scenario, introduce some convenient normalizationconstants E and T, related by E = kB T.

General solution:

T (r1, . . . , rn) =T

f ((r1, . . . , rn)/E);

S(r1, . . . , rn) = kB f ((r1, . . . , rn)/E).

Here f (x) is again an arbitrary monotonic function,and f (x) = df /dx is its derivative.Verify using the chain rule.

Monotonicity of f (x) required to avoid a divide-by-zero error.

Matt Visser (VUW) Conservative entropic forces gtc2011 15 / 55

Conservative entropic forces: n-body

Summary:

T (r1, . . . , rn) =T

f ((r1, . . . , rn)/E);

S(r1, . . . , rn) = kB f ((r1, . . . , rn)/E).

Very simple and very general constraint on the temperature andentropy of any thermodynamic system capable of mimicking thedynamics of n bodies mutually interacting via a conservative force.

Very powerful constraint.

Very problematic for Verlindes proposal.

Matt Visser (VUW) Conservative entropic forces gtc2011 16 / 55

Conservative entropic forces: Newtonian gravity

(r1, , rn) = 1

2

j 6=i

Gmimj|ri rj |

,

T (r1, . . . , rn) =T

f

12E

j 6=i

Gmimj|ri rj |

,

S(r1, . . . , rn) = kB f

12E

j 6=i

Gmimj|ri rj |

.If Newtonian gravity can be mimicked by an entropic force,then, (in view of the monotonicity of f (x)),the entropy must be high when the particles are close together.

Matt Visser (VUW) Conservative entropic forces gtc2011 17 / 55

Conservative entropic forces: Newtonian gravity

Example: A very specific proposal is to take f (x) = x :

T (r1, . . . , rn) = T; S(r1, . . . , rn) =kB2E

j 6=i

Gmimj|ri rj |

.

Simplest possible entropic force model one could come up with forNewtonian gravity.

Certainly reproduces the dynamics of Newtonian gravity.

But very different in detail from Verlindes proposal.

(One reason for possibly being interested in this specific proposal is that itis isothermal, and the known examples of entropic forces in condensedmatter setting typically take place in an isothermal environment.)

Matt Visser (VUW) Conservative entropic forces gtc2011 18 / 55

Conservative entropic forces: Coulomb force

(r1, , rn) =1

80

j 6=i

qiqj|ri rj |

,

T (r1, . . . , rn) =T

f

180 E

j 6=i

qiqj|ri rj |

,

S(r1, . . . , rn) = kB f

180 E

j 6=i

qiqj|ri rj |

.If the Coulomb force can be mimicked by a entropic force,then, (in view of the monotonicity of f (x), and the fact that the Coulombpotential is of indefinite sign),one must be prepared to deal with negative entropies and temperatures.

Matt Visser (VUW) Conservative entropic forces gtc2011 19 / 55

Conservative entropic forces: Coulomb force

Negative entropies and temperatures are outside the realm of classicalthermodynamics, but are nevertheless well-established concepts intheoretical physics.

Negative temperatures are common in statistical physics, where theyare a signal that one is encountering a population inversion.(For example, in certain nuclear spin systems, in certain atomicgasses, or in laser physics.)

Negative entropies are less common, but negentropy is ofteninterpreted in terms of information.(For example Shannons information theory, and various attempts atreinterpreting thermodynamics in terms of information theory.)

Many more instances of negative entropies and negative temperatureswhen we explore Verlindes specific approach.

Matt Visser (VUW) Conservative entropic forces gtc2011 20 / 55

Conservative entropic forces: Coulomb force

Example: A very specific proposal is to take f (x) = x :