Victoria University of Wellington. Te Whare Wānanga o te Ūpoko o te Ika a Maui VUW. Conservative entropic forces. Matt Visser

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1 Victoria University of Wellington Te Whare Wānanga o te Ūpoko o te Ika a Maui VUW Conservative entropic forces Matt Visser Gravity as Thermodynamics: Towards the microscopic origin of geometry ESF Exploratory Workshop SISSA/ISAS, Trieste, Italy Monday 5th September 2011 Matt Visser (VUW) Conservative entropic forces gtc / 55

2 Abstract: VUW Entropic 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 of Verlinde s entropic gravity proposal. Though without directly impacting on either Jacobson s or Padmanabhan s versions of entropic gravity. Resolution? Extend the usual notion of entropic force to multiple heat baths with multiple temperatures and multiple entropies? arxiv: [hep-th]. VUW Matt Visser (VUW) Conservative entropic forces gtc / 55

3 Outline: VUW 1 Background 2 Conservative entropic forces 3 Verlinde s proposal 4 Thermodynamic forces 5 Discussion VUW Matt Visser (VUW) Conservative entropic forces gtc / 55

4 Background: Background Matt Visser (VUW) Conservative entropic forces gtc / 55

5 Background: I shall not attempt to derive or justify an entropic interpretation for Newtonian gravity. Rather I shall ask the converse question: Assuming that Newtonian gravity can be described by an entropic force, what does this tell us about the relevant temperature and entropy functions of the assumed thermodynamic system? Matt Visser (VUW) Conservative entropic forces gtc / 55

6 Background: Start from the definition of an entropic force F = T S. Demand that this entropic force reproduces the conservative force law of Newtonian gravity F = Φ. This places some rather strong constraints on the functional form of the temperature and entropy. Matt Visser (VUW) Conservative entropic forces gtc / 55

7 Conservative entropic forces: Conservative entropic forces Matt Visser (VUW) Conservative entropic forces gtc / 55

8 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 gtc / 55

9 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 specific proposal? 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 r i, for i {1... n}.) Matt Visser (VUW) Conservative entropic forces gtc / 55

10 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 level sets 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 gtc / 55

11 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 = k B T. General solution: T (r) = T f ( Φ(r)/E ) ; S(r) = k B 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. Matt Visser (VUW) Conservative entropic forces gtc / 55

12 Conservative entropic forces: 1-body Summary: T (r) = T f ( Φ(r)/E ) ; S(r) = k B f ( Φ(r)/E ). Very simple and very general constraint on the temperature and entropy of any thermodynamic system capable of mimicking an externally imposed conservative force. Very powerful constraint. Very problematic for Verlinde s proposal. Matt Visser (VUW) Conservative entropic forces gtc / 55

13 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 F i (r 1,..., r n ) = i Φ(r 1,..., r n ). Assume this can be mimicked by an entropic force F i (r 1,..., r n ) = T (r 1,..., r n ) i S(r 1,..., r n ). Implies i Φ(r 1,..., r n ) = T (r 1,..., r n ) i S(r 1,..., r n ). Matt Visser (VUW) Conservative entropic forces gtc / 55

14 Conservative entropic forces: n-body Without any calculation, i we have i Φ i S. Implies that the level sets of the potential are also level sets of the entropy. Implies that the entropy is some function of the potential. Take the curl (with respect to the variable r i ). i we have i T i S. So level sets of the temperature are also level sets of the entropy. Matt Visser (VUW) Conservative entropic forces gtc / 55

15 Conservative entropic forces: n-body As in the 1-body scenario, introduce some convenient normalization constants E and T, related by E = k B T. General solution: T (r 1,..., r n ) = T f ( Φ(r 1,..., r n )/E ) ; S(r 1,..., r n ) = k B f ( Φ(r 1,..., r n )/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 gtc / 55

16 Conservative entropic forces: n-body Summary: T (r 1,..., r n ) = T f ( Φ(r 1,..., r n )/E ) ; S(r 1,..., r n ) = k B f ( Φ(r 1,..., r n )/E ). Very simple and very general constraint on the temperature and entropy of any thermodynamic system capable of mimicking the dynamics of n bodies mutually interacting via a conservative force. Very powerful constraint. Very problematic for Verlinde s proposal. Matt Visser (VUW) Conservative entropic forces gtc / 55

17 Conservative entropic forces: Newtonian gravity Φ(r 1,, r n ) = 1 2 T (r 1,..., r n ) = f S(r 1,..., r n ) = k B f j i Gm i m j r i r j, T 1 2E j i 1 2E j i, Gm i m j r i r j Gm i m j. r i r j 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 gtc / 55

18 Conservative entropic forces: Newtonian gravity Example: A very specific proposal is to take f (x) = x: T (r 1,..., r n ) = T ; S(r 1,..., r n ) = k B Gm i m j 2E r i r j. j i Simplest possible entropic force model one could come up with for Newtonian 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 it is isothermal, and the known examples of entropic forces in condensed matter setting typically take place in an isothermal environment.) Matt Visser (VUW) Conservative entropic forces gtc / 55

19 Conservative entropic forces: Coulomb force Φ(r 1,, r n ) = 1 q i q j 8πɛ 0 r i r j, T (r 1,..., r n ) = f S(r 1,..., r n ) = k B f j i T 1 8πɛ 0 E j i 1 8πɛ 0 E j i, q i q j r i r j q i q j. r i r j 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 Coulomb potential is of indefinite sign), one must be prepared to deal with negative entropies and temperatures. Matt Visser (VUW) Conservative entropic forces gtc / 55

20 Conservative entropic forces: Coulomb force Negative entropies and temperatures are outside the realm of classical thermodynamics, but are nevertheless well-established concepts in theoretical physics. Negative temperatures are common in statistical physics, where they are a signal that one is encountering a population inversion. (For example, in certain nuclear spin systems, in certain atomic gasses, or in laser physics.) Negative entropies are less common, but negentropy is often interpreted in terms of information. (For example Shannon s information theory, and various attempts at reinterpreting thermodynamics in terms of information theory.) Many more instances of negative entropies and negative temperatures when we explore Verlinde s specific approach. Matt Visser (VUW) Conservative entropic forces gtc / 55

21 Conservative entropic forces: Coulomb force Example: A very specific proposal is to take f (x) = x: T (r 1,..., r n ) = T ; S(r 1,..., r n ) = k B q i q j 8πɛ 0 E r i r j. j i Simplest possible entropic force model one could come up with for the Coulomb force. Certainly accurately reproduces the dynamics of the Coulomb force. Qualitatively different from Wang s proposal. At best orthogonal to Verlinde s suggestions. Matt Visser (VUW) Conservative entropic forces gtc / 55

22 Verlinde s proposal: Verlinde s proposal Matt Visser (VUW) Conservative entropic forces gtc / 55

23 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 the Unruh effect. 2 Making the entropy depend on the distance from a holographic screen. We have just seen that for conservative entropic forces we only have one free function f (x) to play with. This is simply not sufficient to satisfy all of Verlinde s requirements. Matt Visser (VUW) Conservative entropic forces gtc / 55

24 Verlinde s proposal: 1-body external potential Verlinde s proposal amounts to: T = a 2πk B c ; S = 2πk Bmc â. That is T = Φ 2πk B mc ; S = 2πk Bmc Φ Φ. But this last equation, S = (const) Φ Φ, is generically ill-posed. Matt Visser (VUW) Conservative entropic forces gtc / 55

25 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 the iso-acceleration surfaces. This is not an argument against entropic forces. Nor even an argument against entropic reinterpretations of Newtonian gravity. It is instead an argument against Verlinde s specific proposals for T and S. Matt Visser (VUW) Conservative entropic forces gtc / 55

26 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 different temperature to each particle T i = a i 2πk B c = iφ, i {1, 2}. 2πk B m i c The standard notion of entropic force really only has room for a single temperature to be assigned to the whole thermodynamic system. Put this aside for now, and concentrate on the entropy... Matt Visser (VUW) Conservative entropic forces gtc / 55

27 Verlinde s proposal: 2-body scenario Verlinde s key axiom is that a particle near a holographic screen in some sense contributes an entropy mc x S = 2πk B. Verlinde takes the entropy to increase as the particle moves towards the holographic screen. Let us call S 0 the entropy of the holographic screen when the particle is located on the screen itself, and l the geodesic distance to the screen. Then at least for small l we can formalize this as S = S 0 2πk B mcl. Matt Visser (VUW) Conservative entropic forces gtc / 55

28 Verlinde s proposal: 2-body scenario Rewrite as: S = 2πk Bmc ˆn. Here ˆn is the outward normal to the holographic screen. The minus sign is important. For two particles we have two masses m i. As long as we are dealing with a central force, in a 2-body system it is appropriate to choose two spherical holographic screens, one around each particle individually, thereby defining two normal vectors ˆn i. Matt Visser (VUW) Conservative entropic forces gtc / 55

29 Verlinde s proposal: 2-body scenario This strongly suggests that we need two entropies i S i = 2πk Bm i c ˆn i, i {1, 2}, (no sum on i). As long as we are dealing with a central force, in a 2-body system ˆn i (r i r not(i) ), i {1, 2}. Because of the very high symmetry, in the 2-body situation we can integrate these two equations: S i = 2πk Bm i c r i r not(i), i {1, 2}. Matt Visser (VUW) Conservative entropic forces gtc / 55

30 Verlinde s proposal: 2-body scenario Note these entropies are negative. Even if we had used the arbitrary constants of integration to make the entropy positive at zero separation, one would nevertheless be driven to negative entropy at large separation. So there is no real loss of generality in choosing to normalize these entropies to zero at zero separation. To reproduce the 2-body force law we must now take F i = T i i S i i {1, 2}, (no sum on i). Matt Visser (VUW) Conservative entropic forces gtc / 55

31 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 the cost of negative entropies S i. (Temperatures T i are positive.) Using two temperatures, and two entropies, to reproduce 2-body Newtonian 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 the 2-body scenario, and dealt exclusively with the test particle limit. Even more restrictively, with the test particle limit in situations of extremely high symmetry. Matt Visser (VUW) Conservative entropic forces gtc / 55

32 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, T i = a i 2πk B c = iφ 2πk B m i c, i {1,..., n}. And with ill-posed differential equations determining the entropies S i. Matt Visser (VUW) Conservative entropic forces gtc / 55

33 Verlinde s proposal: n 3-body scenario At various points of his article, Verlinde rather strongly suggests that his holographic screens be located on equipotential surfaces, in which case the normal is ˆn = Φ/ Φ. But then we are back to the equation S = 2πk Bmc 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.) Matt Visser (VUW) Conservative entropic forces gtc / 55

34 Verlinde s proposal: n 3-body scenario In fact, one should write down one such equation for each individual particle, i S i = 2πk Bm i c i Φ, i {1,, n}, (no sum on i). i Φ But for n 3 bodies the potential Φ(r 1,..., r n ) generically has no symmetries, so these are ill-posed equations that generically have no solutions. We have gone through these problematic issues in some detail because the problems raised now give us some hints on how to proceed. I again emphasize that I am not particularly worried about entropic forces per se, it is instead the combination of entropic forces with the Unruh effect and holographic screens that leads to problems. Matt Visser (VUW) Conservative entropic forces gtc / 55

35 Thermodynamic forces: Thermodynamic forces Matt Visser (VUW) Conservative entropic forces gtc / 55

36 Thermodynamic forces: Complicated thermodynamic system: Described by a large number of intensive variables x a. Correspondingly large number of extensive variables X a. Write down an expression for the thermodynamic force F = a x a X a. More general structure than normally assigned to an entropic force. This decomposition is much more promising when it comes to a coherent implementation of Verlinde s ideas. Matt Visser (VUW) Conservative entropic forces gtc / 55

37 Thermodynamic forces: 1-body external potential We had previously seen that the differential equation determining Verlinde s entropy was ill-posed unless the potential was of very high symmetry. Assume that the potential decomposes into a linear sum of such highly symmetric potentials Φ(r) = a Φ a (r). Let the individual Φ a (r) be either spherically symmetric, cylindrically symmetric, or plane symmetric. Matt Visser (VUW) Conservative entropic forces gtc / 55

38 Thermodynamic forces: 1-body external potential Let l a denote the geodesic distance to the centre of the spherically symmetric potentials, the geodesic distance to the axis of the cylindrically symmetric potentials, and the (signed) geodesic distance to some convenient plane of symmetry for the plane symmetric potentials. Then, by construction, for each individual potential we have Φ a (r) = Φ a (l a ). For each individual potential Φ a we can now integrate S a = 2πk Bmc ˆn a = 2πk Bmc Up to arbitrary irrelevant constants of integration: l a. S a = 2πk Bmc l a. Matt Visser (VUW) Conservative entropic forces gtc / 55

39 Thermodynamic forces: 1-body external potential Define T a = 2πk B c (a a ˆn a ) = 2πk B mc Φ a l a, (no sum on a). As required F = T a S a = a a This works, but... Φ a l a ( ) l a = Φ a = Φ. a Matt Visser (VUW) Conservative entropic forces gtc / 55

40 Thermodynamic forces: 1-body external potential Comments: Thermodynamic interpretation of the force, but with an unboundedly large number of temperatures T a, and entropies S a. Note use of a a ˆn a rather than a a, and Φ a / l a rather than Φ a. Automatically takes care of the signs for attractive and repulsive potentials. Formalism works equally well for gravity and electromagnetism, Can now even handle potentials such as the Lennard Jones potential where the force can change sign as a function of distance. For attractive forces the Unruh-like temperature T a is positive, while for repulsive forces it is negative. Matt Visser (VUW) Conservative entropic forces gtc / 55

41 Thermodynamic forces: 1-body external potential Physical 3-acceleration satisfies a = a a a. Based loosely on the Unruh effect define a total temperature T = a T a ˆn a T a. a The utility of such a definition is uncertain. One might also try to define a total entropy S = a S a = 2πk Bmc The utility of such a definition is uncertain. l a. a Matt Visser (VUW) Conservative entropic forces gtc / 55

42 Thermodynamic forces: n-body scenario General thermodynamic ansatz: Consider any n-body potential that is a linear sum of 2-body central potentials: Φ(r 1,..., r n ) = 1 i j Φ ij (r i r j ). 2 For each ordered pair of particles, based on the 2-body results of the previous section, postulate S i:j = 2πk Bm i c r i r j Note the absence of interchange symmetry. i,j = 2πk Bm i c l ij, i, j {1,..., n}. This is the entropy of particle i due to the presence of particle j. Matt Visser (VUW) Conservative entropic forces gtc / 55

43 Thermodynamic forces: n-body scenario Based very loosely on the Unruh effect, one can argue that there is also a temperature of particle i due to the presence of particle j: T i:j = 2πk B c (a i:j ˆn i:j ) = 2πk B m i c Φ ij l ij, i, j {1,..., n}. Again note the absence of any interchange symmetry. Matt Visser (VUW) Conservative entropic forces gtc / 55

44 Thermodynamic forces: n-body scenario Then F i = j i T i:j i S i:j = j j i j Φ ij l ij ( j i ) ( 1 = i Φ ij = i j 2 j i ˆn i:j = j i i,j j Φ ij ) Φ ij l ij i l ij = i Φ(r 1,..., r n ). This at least reproduces the classical force law we are attempting to emulate using thermodynamic means. Matt Visser (VUW) Conservative entropic forces gtc / 55

45 Thermodynamic forces: n-body scenario To paraphrase Alice, consider the number of impossible things one has to believe in before breakfast: You need a whole collection of n(n 1) temperatures T i:j, one for each ordered pair of particles, which do not add in any sensible way. 3-accelerations of the individual particles now satisfy j i a i = a i:j. So based loosely on the Unruh effect one might guess that each individual particle can be assigned a temperature : j i T i = T i:j ˆn i:j j i T i:j. j But there seems to be no sensible way of defining an overall temperature for the entire n-body system. j j Matt Visser (VUW) Conservative entropic forces gtc / 55

46 Thermodynamic forces: n-body scenario You also need a whole collection of n(n 1) entropies S i:j, one for each ordered pair of particles. Total entropy S: If we boldly assert j i j i S = S i:j = i,j then defining R = max ij { r i r j }, we have i,j 2πk B m i c r i r j, S j i i,j 2πk B m i c R = 2πk BMcR. Up to a sign, this is a Newtonian version of the Bekenstein bound. Whether or not this observation has any deeper significance is unclear. Matt Visser (VUW) Conservative entropic forces gtc / 55

47 Thermodynamic forces: n-body scenario This construction works for any n-body potential that is a linear sum of 2-body central potentials. Both attractive and repulsive forces are automatically dealt with by phrasing the temperatures in terms of l Φ, (rather than l Φ ). Negative entropies. Positive temperatures for attractive forces. Negative temperatures for repulsive forces. Matt Visser (VUW) Conservative entropic forces gtc / 55

48 Thermodynamic forces: n-body scenario It is possible to find an interpretation of Verlinde s ideas that is simultaneously thermodynamic, respects the Unruh-like interpretation of temperature, is compatible with Verlinde s holographic screens, and correctly reproduces the original classical force that one is attempting to emulate. But the price paid for this is very high. Matt Visser (VUW) Conservative entropic forces gtc / 55

49 Thermodynamic forces: Newton and Coulomb forces Newtonian gravity: Take S i:j = 2πk Bm i c l ij, i, j {1,..., n}, and T i:j = 2πk B c Gm j l 2, i, j {1,..., n}. ij Coulomb force: entropies remain the same, but temperatures are modified T i:j = 2πk B m i c q i q j 4πɛ 0 l 2, i, j {1,..., n}. ij Matt Visser (VUW) Conservative entropic forces gtc / 55

50 Thermodynamic forces: Newton and Coulomb forces Though somewhat complicated, this particular assignment of multiple temperatures and entropies seems to be the minimum requirement to make something like Verlinde s suggestions work. Matt Visser (VUW) Conservative entropic forces gtc / 55

51 Discussion: Discussion Matt Visser (VUW) Conservative entropic forces gtc / 55

52 Discussion: I have not attempted to justify reinterpreting Newtonian gravity as an entropic force. I instead I have asked the question: If we assume Newtonian gravity is an entropic force, what does 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. Matt Visser (VUW) Conservative entropic forces gtc / 55

53 Discussion: If we want to use a single heat bath, then any conservative force can be recast into entropic force form but the resulting model is at best orthogonal to Verlinde s proposal. If we wish to retain key parts of Verlinde s proposal (an Unruh-like temperature, and entropy related to holographic screens ), then one is unavoidably forced into a more general thermodynamic force scenario with multiple intensive and extensive thermodynamic variables. Multiple temperatures and entropies. The relevant entropies are negative, while the temperatures are positive for attractive forces and negative for repulsive forces. These features are certainly odd. Certainly not what might naively be expected. Matt Visser (VUW) Conservative entropic forces gtc / 55

54 Discussion: 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, Padmanabhan, and others, there are also good reasons to suspect a thermodynamic interpretation of the fully relativistic Einstein equations might be possible. Whether the specific proposals of Verlinde are anywhere near as fundamental is yet to be seen the rather baroque construction needed to accurately reproduce n-body Newtonian gravity in a Verlinde-like setting certainly gives one pause. Matt Visser (VUW) Conservative entropic forces gtc / 55

55 End: VUW Matt Visser (VUW) Conservative entropic forces gtc / 55