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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

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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 ofVerlinde’s entropic gravity proposal.

Though without directly impacting on either Jacobson’s orPadmanabhan’s 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].

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Outline:

VUW1 Background

2 Conservative entropic forces

3 Verlinde’s proposal

4 Thermodynamic forces

5 Discussion

VUW

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Background:

Background

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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?

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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 = −∇Φ.

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

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Conservative entropic forces:

Conservative entropic forces

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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.

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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 Verlinde’s 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}.)

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Conservative entropic forces: 1-body

Single body interacting with an externally specified potential

F(r) = −∇Φ(r).

Assume this can be mimicked by an entropic force

F(r) = T (r) ∇S(r).

Implies∇Φ(r) = −T (r) ∇S(r).

Without any calculation, since ∇Φ||∇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.

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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 ′(−Φ(r)/E∗); S(r) = kB f (−Φ(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.

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Conservative entropic forces: 1-body

Summary:

T (r) =T∗

f ′(−Φ(r)/E∗); S(r) = kB f (−Φ(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 Verlinde’s proposal.

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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 Φ(r1, . . . , rn).

Assume this can be mimicked by an entropic force

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

Implies

∇i Φ(r1, . . . , rn) = −T (r1, . . . , rn) ∇iS(r1, . . . , rn).

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Conservative entropic forces: n-body

Without any calculation, ∀i we have ∇i Φ||∇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.

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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 ′(−Φ(r1, . . . , rn)/E∗);

S(r1, . . . , rn) = kB f (−Φ(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.

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Conservative entropic forces: n-body

Summary:

T (r1, . . . , rn) =T∗

f ′(−Φ(r1, . . . , rn)/E∗);

S(r1, . . . , rn) = kB f (−Φ(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 Verlinde’s proposal.

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Conservative entropic forces: Newtonian gravity

Φ(r1, · · · , rn) = −1

2

∑j 6=i

Gmimj

|ri − rj |,

T (r1, . . . , rn) =T∗

f ′

1

2E∗

∑j 6=i

Gmimj

|ri − rj |

,

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

1

2E∗

∑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.

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Conservative entropic forces: Newtonian gravity

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

T (r1, . . . , rn) = T∗; S(r1, . . . , rn) =kB

2E∗

∑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 Verlinde’s 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.)

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Conservative entropic forces: Coulomb force

Φ(r1, · · · , rn) =1

8πε0

∑j 6=i

qiqj

|ri − rj |,

T (r1, . . . , rn) =T∗

f ′

− 1

8πε0 E∗

∑j 6=i

qiqj

|ri − rj |

,

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

− 1

8πε0 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.

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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 Shannon’s information theory, and various attempts atreinterpreting thermodynamics in terms of information theory.)

Many more instances of negative entropies and negative temperatureswhen we explore Verlinde’s specific approach.

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Conservative entropic forces: Coulomb force

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

T (r1, . . . , rn) = T∗; S(r1, . . . , rn) = − kB

8πε0 E∗

∑j 6=i

qiqj

|ri − rj |.

Simplest possible entropic force model one could come up with forthe Coulomb force.

Certainly accurately reproduces the dynamics of the Coulomb force.

Qualitatively different from Wang’s proposal.

At best orthogonal to Verlinde’s suggestions.

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Verlinde’s proposal:

Verlinde’s proposal

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Verlinde’s proposal:

The problems with Verlinde’s proposal are two-fold.

They come from his specific suggestions for:

1 Making the temperature depend on a non-relativistic variant of theUnruh effect.

2 Making the entropy depend on the “distance from a holographicscreen”.

We have just seen that for conservative entropic forces we only have onefree function f (x) to play with.

This is simply not sufficient to satisfy all of Verlinde’s requirements.

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Verlinde’s proposal: 1-body external potential

Verlinde’s proposal amounts to:

T =~ |a|

2πkBc; ∇S =

2πkBmc

~a.

That is

T =~ |∇Φ|

2πkBmc; ∇S = −2πkBmc

~∇Φ

|∇Φ|.

But this last equation,

∇S = (const)∇Φ

|∇Φ|,

is generically ill-posed.

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Verlinde’s proposal: 1-body external potential

∇S = (const)∇Φ

|∇Φ|.

It is only when the level sets of Φ coincide with the level sets of |∇Φ|that this differential equation has solutions.

That is, the iso-potential surfaces have to coincide with theiso-acceleration surfaces.

This is not an argument against entropic forces.

Nor even an argument against entropic reinterpretations of Newtoniangravity.

It is instead an argument against Verlinde’s specific proposals for Tand ∇S .

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Verlinde’s proposal: 2-body scenario

Somewhat different problems affect the 2-body scenario.

At the most basic level Verlinde’s proposal would assign a differenttemperature to each particle

Ti =~ |ai |

2πkBc=

~ |∇i Φ|2πkBmic

, i ∈ {1, 2}.

The standard notion of entropic force really only has room for a singletemperature to be assigned to the whole thermodynamic system.

Put this aside for now, and concentrate on the entropy...

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Verlinde’s proposal: 2-body scenario

Verlinde’s key axiom is that a particle near a “holographic screen” insome sense contributes an entropy

∆S = 2πkBmc∆x

~.

Verlinde takes the entropy to increase as the particle moves towardsthe “holographic screen”.

Let us call S0 the entropy of the “holographic screen” when theparticle is located on the screen itself, and ` the geodesic distance tothe screen.

Then at least for small ` we can formalize this as

S = S0 − 2πkBmc`

~.

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Verlinde’s proposal: 2-body scenario

Rewrite as:

∇S = −2πkBmc

~n.

Here n is the “outward normal to the holographic screen”.

The minus sign is important.

For two particles we have two masses mi .

As long as we are dealing with a central force, in a 2-body system it isappropriate to choose two spherical “holographic screens”, one aroundeach particle individually, thereby defining two normal vectors ni .

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Verlinde’s proposal: 2-body scenario

This strongly suggests that we need two entropies

∇iSi = −2πkBmic

~ni , i ∈ {1, 2}, (no sum on i).

As long as we are dealing with a central force, in a 2-body system

ni || (ri − rnot(i)), i ∈ {1, 2}.

Because of the very high symmetry, in the 2-body situation we canintegrate these two equations:

Si = −2πkBmic

~|ri − rnot(i)|, i ∈ {1, 2}.

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Verlinde’s proposal: 2-body scenario

Note these entropies are negative.

Even if we had used the arbitrary constants of integration to make theentropy positive at zero separation, one would nevertheless be drivento negative entropy at large separation.

So there is no real loss of generality in choosing to normalize theseentropies to zero at zero separation.

To reproduce the 2-body force law we must now take

Fi = Ti ∇iSi i ∈ {1, 2}, (no sum on i).

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Verlinde’s proposal: 2-body scenario

But there are various ways in which this proposal still does not quite work.

Newtonian gravity: This generates an attractive 2-body force, at thecost of negative entropies Si . (Temperatures Ti are positive.)

Using two temperatures, and two entropies, to reproduce 2-bodyNewtonian gravity is orthogonal to standard notions of entropic force.

Coulomb 2-body situation:Additional ad hoc “fix” to keep track of attraction versus repulsion.

Same sign charges (repulsive forces) need negative temperatures.

Why unrecognized in Verlinde’s article?

Because the explicit calculations carried out there did not look at the2-body scenario, and dealt exclusively with the test particle limit.

Even more restrictively, with the test particle limit in situations ofextremely high symmetry.

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Verlinde’s proposal: n ≥ 3-body scenario

Related but even more acute problems affect the n-body scenario.

For n ≥ 3 one has to deal:

Both with multiple temperatures,

Ti =~ |ai |

2πkBc=

~ |∇i Φ|2πkBmic

, i ∈ {1, . . . , n}.

And with ill-posed differential equations determining the entropies Si .

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Verlinde’s proposal: n ≥ 3-body scenario

At various points of his article, Verlinde rather strongly suggests thathis “holographic screens” be located on equipotential surfaces, inwhich case the normal is n = ∇Φ/|∇Φ|.But then we are back to the equation

∇S = −2πkBmc

~∇Φ

|∇Φ|,

which we had previously seen is generically ill-posed.

(That is, ill-posed except in situations of extremely high symmetry.)

(Spherical, cylindrical, or planar symmetry.)

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Verlinde’s proposal: n ≥ 3-body scenario

In fact, one should write down one such equation for each individualparticle,

∇iSi = −2πkBmic

~∇i Φ

|∇i Φ|, i ∈ {1, · · · , n}, (no sum on i).

But for n ≥ 3 bodies the potential Φ(r1, . . . , rn) generically has nosymmetries, so these are ill-posed equations that generically have nosolutions.

We have gone through these problematic issues in some detail becausethe problems raised now give us some hints on how to proceed.

I again emphasize that I am not particularly worried about entropicforces per se, it is instead the combination of entropic forces with theUnruh effect and “holographic screens” that leads to problems.

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Thermodynamic forces:

Thermodynamic forces

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Thermodynamic forces:

Complicated thermodynamic system:

Described by a large number of intensive variables xa.Correspondingly large number of extensive variables Xa.

Write down an expression for the “thermodynamic force”

F =∑

a

xa ∇Xa.

More general structure than normally assigned to an “entropic force”.

This decomposition is much more promising when it comes to acoherent implementation of Verlinde’s ideas.

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Thermodynamic forces: 1-body external potential

We had previously seen that the differential equation determiningVerlinde’s entropy was ill-posed unless the potential was of very highsymmetry.

Assume that the potential decomposes into a linear sum of suchhighly symmetric potentials

Φ(r) =∑

a

Φa(r).

Let the individual Φa(r) be either spherically symmetric, cylindricallysymmetric, or plane symmetric.

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Thermodynamic forces: 1-body external potential

Let `a denote the geodesic distance to the centre of the sphericallysymmetric potentials, the geodesic distance to the axis of thecylindrically symmetric potentials, and the (signed) geodesic distanceto some convenient plane of symmetry for the plane symmetricpotentials.

Then, by construction, for each individual potential we haveΦa(r) = Φa(`a).

For each individual potential Φa we can now integrate

∇Sa = −2πkBmc

~na = −2πkBmc

~∇`a.

Up to arbitrary irrelevant constants of integration:

Sa = −2πkBmc

~`a.

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Thermodynamic forces: 1-body external potential

Define

Ta = − ~2πkBc

(aa · na) =~

2πkBmc

∂Φa

∂`a, (no sum on a).

As required

F =∑

a

Ta ∇Sa = −∑

a

∂Φa

∂`a∇`a = −∇

(∑a

Φa

)= −∇Φ.

This “works”, but...

Matt Visser (VUW) Conservative entropic forces gtc2011 39 / 55

Thermodynamic forces: 1-body external potential

Comments:

Thermodynamic interpretation of the force, but with an unboundedlylarge number of “temperatures” Ta, and “entropies” Sa.

Note use of aa · na rather than |aa|, and ∂Φa/∂`a rather than ∇Φa.

Automatically takes care of the signs for attractive and repulsivepotentials.

Formalism works equally well for gravity and electromagnetism,

Can now even handle potentials such as the Lennard–Jones potentialwhere the force can change sign as a function of distance.

For attractive forces the Unruh-like temperature Ta is positive,while for repulsive forces it is negative.

Matt Visser (VUW) Conservative entropic forces gtc2011 40 / 55

Thermodynamic forces: 1-body external potential

Physical 3-acceleration satisfies

a =∑

a

aa.

Based loosely on the Unruh effect define a “total temperature”

T =∣∣∣∑

aTa na

∣∣∣ ≤∑a|Ta| .

The utility of such a definition is uncertain.

One might also try to define a “total entropy”

S =∑

a

Sa = −2πkBmc

~∑

a

`a.

The utility of such a definition is uncertain.

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Thermodynamic forces: n-body scenario

General “thermodynamic” ansatz:

Consider any n-body potential that is a linear sum of 2-body centralpotentials:

Φ(r1, . . . , rn) =1

2

i 6=j∑i ,j

Φij (ri − rj ).

For each ordered pair of particles, based on the 2-body results of theprevious section, postulate

Si :j = −2πkBmic |ri − rj |

~= −

2πkBmic `ij

~, i , j ∈ {1, . . . , n}.

Note the absence of interchange symmetry.

This is the “entropy” of particle i due to the presence of particle j .

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Thermodynamic forces: n-body scenario

Based very loosely on the Unruh effect, one can argue that there isalso a “temperature” of particle i due to the presence of particle j :

Ti :j = − ~2πkBc

(ai :j · ni :j ) =~

2πkBmic

∂Φij

∂`ij, i , j ∈ {1, . . . , n}.

Again note the absence of any interchange symmetry.

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Thermodynamic forces: n-body scenario

Then

Fi =

j 6=i∑j

Ti :j ∇iSi :j = −j 6=i∑

j

∂Φij

∂`ijni :j = −

j 6=i∑j

∂Φij

∂`ij∇i`ij

= −∇i

(∑j 6=i

jΦij

)= −∇i

(1

2

∑j 6=i

i ,jΦij

)= −∇i Φ(r1, . . . , rn).

This at least reproduces the classical force law we are attempting toemulate using thermodynamic means.

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Thermodynamic forces: n-body scenario

To paraphrase Alice, consider the number of impossible things one has tobelieve in before breakfast:

You need a whole collection of n(n − 1) “temperatures” Ti :j , one foreach ordered pair of particles, which do not add in any sensible way.

3-accelerations of the individual particles now satisfy

ai =

j 6=i∑j

ai :j .

So based loosely on the Unruh effect one might guess that eachindividual particle can be assigned a “temperature”:

Ti =

∣∣∣∣∑j 6=i

jTi :j ni :j

∣∣∣∣ ≤∑j 6=i

j|Ti :j |.

But there seems to be no sensible way of defining an overall“temperature” for the entire n-body system.

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Thermodynamic forces: n-body scenario

You also need a whole collection of n(n − 1) entropies Si :j , one foreach ordered pair of particles.

Total entropy S :If we boldly assert

S =

j 6=i∑i,j

Si :j = −j 6=i∑i,j

2πkBmic |ri − rj |~

,

then defining R = maxij{ |ri − rj |}, we have

|S | ≤j 6=i∑i,j

2πkBmic R

~=

2πkBMcR

~.

Up to a sign, this is a Newtonian version of the Bekenstein bound.Whether or not this observation has any deeper significance is unclear.

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Thermodynamic forces: n-body scenario

This construction works for any n-body potential that is a linear sumof 2-body central potentials.

Both attractive and repulsive forces are automatically dealt with byphrasing the “temperatures” in terms of ∂`Φ, (rather than |∂`Φ|).

Negative entropies.

Positive temperatures for attractive forces.

Negative temperatures for repulsive forces.

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Thermodynamic forces: n-body scenario

It is possible to find an interpretation of Verlinde’s ideas that issimultaneously “thermodynamic”, respects the Unruh-likeinterpretation of temperature, is compatible with Verlinde’s“holographic screens”, and correctly reproduces the original classicalforce that one is attempting to emulate.

But the price paid for this is very high.

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Thermodynamic forces: Newton and Coulomb forces

Newtonian gravity:Take

Si :j = −2πkBmic `ij

~, i , j ∈ {1, . . . , n},

and

Ti :j =~

2πkBc

Gmj

`2ij, i , j ∈ {1, . . . , n}.

Coulomb force:“entropies” remain the same, but “temperatures” are modified

Ti :j = − ~2πkBmic

qiqj

4πε0 `2ij, i , j ∈ {1, . . . , n}.

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Thermodynamic forces: Newton and Coulomb forces

Though somewhat complicated, this particular assignment of multiple“temperatures” and “entropies” seems to be the minimumrequirement to make something like Verlinde’s suggestions work.

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Discussion:

Discussion

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Discussion:

I have not attempted to justify reinterpreting Newtonian gravity as anentropic force.

I instead I have asked the question:

“If we assume Newtonian gravity is an entropic force, whatdoes this tell us about the relevant thermodynamic system?”

What can we say about the temperature and entropy functions?

What constraints do they satisfy?

The answers we have obtained are mixed.

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Discussion:

If we want to use a single heat bath, then any conservative force canbe recast into entropic force form — but the resulting model is atbest orthogonal to Verlinde’s proposal.

If we wish to retain key parts of Verlinde’s proposal (an Unruh-liketemperature, and entropy related to “holographic screens”), then oneis unavoidably forced into a more general “thermodynamic force”scenario with multiple intensive and extensive thermodynamicvariables.

Multiple “temperatures” and “entropies”.The relevant “entropies” are negative, while the “temperatures” arepositive for attractive forces and negative for repulsive forces.These features are certainly odd.Certainly not what might naively be expected.

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Discussion:

There is no reasonable doubt concerning the physical reality ofentropic forces, and no reasonable doubt that classical (andsemi-classical) general relativity is closely related to thermodynamics.

Based on the work of Jacobson, Padmanabhan, and others, there arealso good reasons to suspect a thermodynamic interpretation of thefully relativistic Einstein equations might be possible.

Whether the specific proposals of Verlinde are anywhere near asfundamental is yet to be seen — the rather baroque constructionneeded to accurately reproduce n-body Newtonian gravity in aVerlinde-like setting certainly gives one pause.

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End:

VUW

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