When long-range interactions are present the usual definition of temperature implies that two systems in thermal equilibrium can be at different temperatures. This local temperature has physical significance, if the sub-systems cease to interact, each system will be at their different local temperatures. This is formally related to redshifting of temperature in general relativity. We propose experiments to test this effect which are feasible using current microfabrication techniques. It is also possible to display thermodynamical analogues to black-hole space-time

K. L. Oi, Daniel

Oppenheim, Jonathan

English

arXiv

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Equilibrium temperature anisotropy and black-hole analogues

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryK. L. Oi, Daniel and Oppenheim, Jonathan (2007) Equilibrium temperature anisotropy and black-hole analogues. Working paper. UNSPECIFIED.Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/arXiv:gr-qc/0702010v1  1 Feb 2007Equilibrium temperature anisotropy and black-hole analoguesDaniel K. L. Oi(1)(2), Jonathan Oppenheim(1)(1)Department of Applied Mathematics and Theoretical Physics,Wilberforce Road, University of Cambridge, Cambridge CB3 0WA, UK and(2)SUPA, Department of Physics, University of Strathclyde,Glasgow G4 0NG, UKWhen long-range interactions are present the usual definition of temperature implies that twosystems in thermal equilibrium can be at different temperatures. This local temperature has phys-ical significance, if the sub-systems cease to interact, each system will be at their different localtemperatures. This is formally related to redshifting of temperature in general relativity. We pro-pose experiments to test this effect which are feasible using current microfabrication techniques. Itis also possible to display thermodynamical analogues to black-hole space-time.INTRODUCTIONWhen a system possesses long-range interactions,many familiar notions of thermodynamics break down;the micro-canonical and canonical ensembles become in-equivalent [1], there may be no stable equilibrium config-uration [2, 3], and heat-capacities can be negative [3, 4]as observed in fragmenting nuclei [5] and atomic clus-ters [6]. Formally, standard thermodynamics has onlybeen proved valid when interactions are short-range, orwhen the long-range interactions are screened (e.g. plas-mas) [7]. There are few methods for studying the ther-modynamics of systems with long-range interactions, al-though some models have been studied for special caseswhere the thermodynamic limit exists [8], and under-standing them outside the standard framework usingTsallis or Renyi entropy has been attempted.Using a general formalism for studying such systems [9]based on techniques used in general relativity (where theequivalence principle is exploited) – an effect was notedwhich we summarise and extend as follows. Considermany strongly interacting sub-systems in thermal equi-librium. Using the standard definition of temperature(defined as the local temperature), each sub-system is ata different temperature even though the entire system isat thermal equilibrium [9]. Clearly the standard defini-tion does not satisfy a basic notion – that it be constantthroughout the sample at equilibrium, yet it has a physi-cal meaning – if the interaction is turned off suddenly andthe sub-systems isolated, they will be at their local tem-peratures. The observed temperature difference when asystem is broken down into its parts is a property of thesystem and is a function of its self-interaction. What’smore, the effect is formally related to effects found incurved space and black hole thermodynamics [9]. Here weshow that this anisotropy of local temperature can be ob-served, though this requires the long-range interaction tobe strong compared with the characteristic temperatureof the sample. However, recent advances in microfabrica-tion may allow experimental access to thermodynamicaleffects not found in macroscopic systems. An experimentto measure the break-down of temperature in quantumsystems was recently proposed in [10].We review the basic formalism for analysing systemswith long range interactions and then apply this to a pro-posed experiment. We also show that one can observeadditional effects closely related to thermodynamical re-lations in black-hole space-times.LONG-RANGE INTERACTIONSConsider two sub-systems, with total energy m of theformm = E1 + E2 +G(E1, E2), (1)where Ei are the local (non-interacting) energies andG(E1, E2) is some interaction potential (which may in-clude self-interacting terms). As an example, we considerthe interaction of two clusters of classical spins in localmagnetic fields b1 and b2, with spin-spin interactionm = b1e1 + b2e2 − J1e21/2− J2e22/2− J12e1e2, (2)where e1 and e2 are the spin-excesses of each cluster ei =n↑i − n↓i , (n↑i and n↓i being the number of up and downspins respectively of cluster i = 1, 2), Ji are intra-clustercouplings, and J12 is the inter-cluster coupling. Such aninteraction arises from a standard interaction betweenspins of the formm =∑b1σj1 +∑b2σk2 −∑lm∑i,i′=1,2Jklii′σki σ′li (3)with the spins of cluster i being σji . For small clusters, theinteraction within each cluster can be approximately thesame for all spins, not just nearest neighbour, i.e. Jkl11 =J1, Jkl22 = J2, and the interaction between each cluster isapproximately uniform over each cluster i.e. Jkl12 = J12.We then drop constant terms to obtain Eq. (2). Gravityis another common example of such an interaction, wherethe potential term depends on the local energies (masses)of each system.One can define the temperature of one of the clusters(the system S), by making the other system very large2(the reservoir R). If the total energy m is fixed, in thislimit the probability that the system has energy ES isgiven by [9]p(ES)dES =∑EΩS(ES)ΩR(E)e−ESβEdES/Zm (4)where the sum is taken over all E = ER +ES consistentwith total energy m = E +G(ES , ER) andZm =∫mdESΩS(ES)eSR(ER) (5)i.e. Zm is the total number of states at fixed total energym. We then define the inverse temperature βE in theusual manner in terms of the local extensive entropyβE ≡∂SR(ER)∂ER(6)We shall refer to βE as the local temperature. The moti-vation for using this term comes from general relativity.Note that the temperature of the system is definedin terms of the derivative of the reservoir’s entropy. Inthe non-interacting case, no issues arise from this defini-tion: if two systems are in thermal contact in the micro-canonical ensemble, then ∂SS(ES)∂ES≃ ∂SR(E)∂E. This is alsotrue when the reservoir has no long-range interactions,or when the division of a single system into a reservoirand smaller system is purely formal (as we will see fromsymmetry considerations). In general however, this is notnecessarily true – a point which will be discussed shortly.One therefore should keep in mind that the temperatureis a property of a reservoir – it gives the distribution asso-ciated with a smaller system in contact with it. Finally,note that the local temperature, as defined, is a functionof E, we make this explicit by writing βE . There willbe different “temperatures” depending on the value of Ethe entire system is found in.Let us now show that at equilibrium, a system can havean anisotropy in local temperature. Here we considera slightly different situation and derivation than in [9].For concreteness, one could consider a system, as above,composed of two clusters as above labelled 1 and 2. Forfixed E1 and E2, the spin excesses of each cluster arefixed, thus their respective entropy is also fixed and givenbySi(Ei) = −n↑i logn↑in↑i + n↓i− n↓i logn↓in↑i + n↓i(7)and since each entropy is determined only by the spinexcess, the total entropy is additive, i.e. the number ofstates accessible to two systems is given by the numberof states accessible to one system, times the number ofstates accessible to the other. I.e.Ω12(E1, E2) = Ω1(E1)Ω2(E2) (8)where Ω12 is the total number of states of the com-bined system, and Ωi is the number of states accessi-ble to system i (the entropies of each system are justSi(Ei) = logΩi(Ei)). Additivity of entropy holds be-cause we will find that we only need it at each value ofE1, E2 and in this case, the entropy is additive in manysystems because often the only correlations in a systemare related to correlations in energy. We can thus con-sider more general systems with additive entropies (whenconditioned on local energy). We now consider these twosystems in contact with a reservoir R at fixed total en-ergy m = E1 + E2 + ER + G(E1, E2) and again tak-ing the total local energy as E = E1 + E2 + ER. Theprobability that the two systems have energy E1 and E2for given local energy E is Ω1(E1)Ω2(E2)ΩR(ER)/ZE ,and the probability that the local energy is E is de-noted by p(E) = ZE/Zm, with the partition functionsZm =∑E1,E2,EΩ1(E1)Ω2(E2)ΩR(ER) at fixed m andZE =∑E1,E2Ω1(E1)Ω2(E2)ΩR(ER) at fixed E. Thenthe probability that system 1 has energy E1 isp1(E1) =∑E,E2p(E1, E2|E)ZE/Zm=∑E,E2Ω1(E1)Ω2(E2)ΩR(ER)/Zm (9)≃Ω1(E1)ZmeSR(m)−βoE1∑E,E2Ω2(E2)e−βo(E2+G(E1,E2))where we have used the approximation that ER ≫E1 + E2 + G(E1, E2) to expand SR around m with theinverse global temperature defined as βo ≡ ∂SR(m)/∂m.Sincem is held fixed, an absence of interaction G(E1, E2)would give the usual fact that p1(E1) will be proportionalto Ω1(E1)e−βoE1 , and symmetrically for p2(E2) – thusthe systems will be at equal temperature. Here, we havethe additional sum over E and the factor due to the in-teraction term, which makes the distribution of system 1and system 2 asymmetric. Expanding G(E1, E2) around(E1, E2) = (E¯1, E¯2) the average energies, givesp(E1) ∝ Ω1(E1)e−(1+∂G(E¯1,E¯2)∂E¯1)βoE1 (10)which looks like a canonical distribution. By symmetry,each system thus behaves as if it has an inverse temper-atureβi = (1 +∂G(E¯1, E¯2)∂E¯i)βo (11)One can calculate that this local temperature matchesthe standard definition of temperature given by Eq. (6)and we thus see that there is a temperature anisotropywhenever ∂G(E¯1, E¯2)/∂E¯2 6= ∂G(E¯1, E¯2)/∂E¯1Let us now understand the physical meaning of thislocal temperature. Eq. (4) gives the probability dis-tribution of a system S in terms of the local temper-ature β, and the local energy ES . However, the lo-cal energy is not a conserved quantity, and does not3contain the interacting term. To see what the locallyconserved energy is, we can expand m to first orderm(E1, E2) ≃ m(E¯1, E¯2)+∂m∂E1(E1− E¯1)+∂m∂E2(E2− E¯2).Since constant terms can be ignored, Eio ≡∂m∂EiEi can beidentified with the energy of each system in the presenceof the mean field due to the other interacting systemsand is (1 + ∂G(E¯1,E¯2)∂E¯i)Ei. In the case of a single systemand reservoir, this energy is Eo ≡ ES∂m∂ESand serves totell us how we should define the locally conserved energy.As is expected for a non-extensive system, the quantityis also non-additive, but, as with the local temperature,will correspond with what happens in general relativity.Eo is also a function of E but we will not write thisexplicitly. With respect to energy levels Eo, each sys-tem acts approximately as if it is at inverse temperatureβo (although in actual fact it is a mixture of tempera-tures [9]). We can think of Eo as the effective energy,i.e. it is the energy of the system in the presence of in-teraction with another system, and βo can be thought ofas the closest thing one has to a physical temperature –looking at Eq. (11) we see that the global inverse temper-ature βo will be equal for both systems. Note however,that the global temperature is not an intensive quantity.If the interaction term is suddenly “turned off”, (onecan imagine that the spins are suddenly separated sothat they no longer interact), then the energy that isascribed to each energy level is no longer Eio, but ratherits local energy E1 or E2. If the change happens ex-tremely quickly then the overall state of the system willnot change. SinceβiEi = βoEio (12)and the new energy of each system is now Ei, the mea-sured temperatures will be βi – the local temperature. Inthese systems, the local temperature becomes the physi-cal one when the interaction is turned off, which is anal-ogous to the fact that in general relativity, the local tem-perature is the physical one measured by free-falling ob-servers (for whom the gravitational interaction is “turnedoff”).Using the two coupled Ising models of Eq. (2) we wouldget a temperature difference ofβ2 = β11− J12e1/b2 − J2e2/b21− J12e2/b1 − J1e1/b1. (13)Now if initially these two systems (or clusters), arefar apart, and at equal global temperature, one cannotpush them together both adiabatically and isothermally(constant global temperature) as is possible in the non-interacting case. This can be seen from Eq. (12). Movingthe systems together adiabatically requires keeping Eoβofixed, but since Eo changes when J12 becomes significant,one cannot keep βo constant. By recalculating Eo one cantherefore calculate the new global temperature. We seetherefore that the global temperature is not an intensivequantity.BLACK-HOLE ON A BENCHTOPTo obtain an analogue of a black-hole, we consider asingle cluster of spins with long-range self-interaction.This can be obtained by setting h2 = J2 = J12 = 0. Thisgives a relation between the local and global temperatureofβ = ±βo√1−2J2mh2. (14)The positive solution looks exactly like the relation-ship between global and local temperature in theSchwarzschild black-hole space-time due to redshifting,if we set the radius r = h2 and equate the spin-spincoupling with Newton’s constant J1 = G. Here βo isthe temperature at infinity, while β would be the localtemperature measured by an observer sitting close to theblack-hole horizon. In the analogue, the horizon is real –for fixed m, there is no value of spin-excess which allowsr < 2Gm i.e. m > h2/2J1. There are also many similar-ities between this analogue and a black hole in terms ofthe way the entropy and energy behave, and we refer thereader to [9, 11] for details.EXPERIMENTAL REALISATIONIdeally, we would like to study these phenomena incomplex macroscopic systems. However these usuallydo not possess a strong enough long-range interactionto produce an appreciable effect, hence it is necessary togo to small systems where the the coupling between sub-systems can be comparable to the local energies and lowtemperatures are easier to achieve.The simplest system to study that of two spin clusterscoupled via an Ising-like interaction,H = ~b1 · ~σ1 +~b2 · ~σ2 +J2σ1z ⊗ σ2z (15)where ~bi are local magnetic fields and J is the interac-tion energy between the two clusters. We can map thissystem to a pair of pseudo-spins such as double quantumdots with a excess electrons whose localisation in one orthe other dot defines the pseudo-spin vector (Fig. 1 orFig. 2). A single electron transistor measures the totalcharge excess in one dot or the other.To implement the separation of the two sub-systemsbefore measurement, the device in Fig. 2 could be used.A surface acoustic wave (SAW), produced in a piezoelec-tric material by an inter-digital transducer driven by amicrowave generator, induces a travelling sinusoidal elec-tric field in a 2-dimensional electron gas (2DEG) situ-ated just below the surface of a modulation-doped GaAs-AlGaAs semiconductor. Bias electrodes on the surface ofthe device deplete the 2DEG of conduction electrons onthe region of the system.4SETTunnelingGateBiasGateJ-CouplingeeFIG. 1: Coupled double quantum dot (DQD) system con-structed from donor phosphorus atoms in silicon with 20nmvertical and 30nm lateral separation. Each left and rightDQD pair is initialised with one electron shared between thetwo dots. The local Hamiltonians can be controlled by elec-trodes affecting the tunnelling rate and bias between the twodots. The interaction is of the Ising type (J = 2.9meV ) andsingle electron transistors (SETs) measure the localisation ofthe electrons to the upper or lower dots of each pair.By suitable electrode potentials at the entrance of thesystem, an exact number of electrons can be transportedin moving quantum dots, along quasi-1D quantum chan-nels (defined by surface electrodes) through the device.Double well potentials can be created with a specifiednumber of electrons in each, and by using suitable elec-trode geometry, made to interact via an Ising coupling.Travelling along a sufficiently long channel, the sub-systems are allowed to thermalise, after which the twosub-systems are rapidly separated and read-out by sim-ply measuring the current via Ohmic contacts.JO acknowledges the support of the Royal Society,and EU grants QAP and COSLAB. DKLO acknowledgesthe CMI programme on quantum information, Fujitsu,EU grants TOPQIP (IST-2001-39215), RESQ (IST-2001-37559), Sidney Sussex College, Cambridge, QIPIRC (EP-SRC UK), and SUPA.[1] For a good review, see for example T. Padmanabhan,Phys Rep, 188, 285-362 (1990).[2] V. A. Antanov, Vest. Leningr. Gos. Univ: Mat. Mekh.Astron. 7, 135 (1962).[3] D. Lynden-Bell and R. Wood, MNRAS 128, 495 (1968).[4] W. Thirring, Z. Phys 235, 339 (1970).[5] M. A. et. al., Phys. Lett. B 473, 219 (2000).[6] M. Schmidt, R. Kusche, T. Hippler, J. Donges, W. Kron-muller, B. von Issendorff, and H. Haberland, Phys. Rev.InitialisationThermalisationMeasurementOhmicContactsJ-CouplingTunnelingBiasDVa) Potential Cross-SectionElectronSAW PropagationaSurfaceElectrodesGaAsGaAsAlGaAs UndopedAlGaAs Doped2D Electron Gas(2DEG)SAW DeviceVerticalCross-SectionSAW Transducer Surface ElectrodesSAW TransportedQuantum WellsFIG. 2: Surface acoustic wave (SAW) device. The SAW cre-ated by an interdigital transducer produces a sinusoidal piezo-electric field in the direction of propagation which, togetherwith surface electrode located above a 2DEG, create a patternof moving quantum wells which can be manipulated by staticvoltages. The advantage of this configuration is that the in-teracting sub-systems can be separated before measurementof their respective populations. Correlated current readoutmeasures the population in the position basis.2040608010012014020 40 60 80Global Temp. (K)ClusterTemp.(K)Cluster 1Cluster 2FIG. 3: A plot of the temperatures of two interactingsub-systems consisting of several electrons with Coulomb ex-change energy.Lett. 86, 1191 (2001).[7] J. Lebowitz and E. Lieb, Phys. Rev. Lett. 22, 631 (1969).[8] F. Dyson, Commun. Math. Phys. 12, 91 (1969).[9] J. Oppenheim, Phys. Rev. E 68, 016108 (2003), gr-qc/0212066.[10] H. Hartmann and G. Mahler, Europhys. Lett 70, 579(2005).[11] J. Oppenheim, Phys. Rev. D 65, 024020 (2002), gr-qc/0105101.

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Equilibrium temperature anisotropy and black-hole analogues

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