Conjecture of new inequalities for some selected thermophysical properties values

In 2005 it was rigorously shown with string theory methods that there is a lower bound for the ratio of the shear viscosity η and the volume density of entropy s = S/V given by η / s ≥ ℏ / ( 4 π k B ) . Here we extend this result in a heuristic manner to other ratios of thermophysical properties. We conjectured that there are rigorous non-zero lower bounds for the Lorenz number L as well as other combinations of equilibrium and transport properties. We suggest that the lower bounds and the corresponding inequalities can be written in terms of the Planck units. We show that some of the proposed new inequalities set severe constraints on the behavior and properties of ordinary matter.


Introduction
In 2005 Kovtun, Son, and Starinets (KSS) [1] studied the shear viscosity in strongly interacting quantum field theories. These studies are of considerable importance in black hole and high-energy physics for which one example is the quark-gluon plasma produced in heavy-ion collisions, see e.g. [2,3]. In particular KSS concentrate on the ratio κ=η/s of the shear viscosity η to the volume density of entropy s=S/V. They κcannot become smaller than κ 0 which sets a rigorous lower bound on this ratio. Since its discovery numerous papers in high-energy and black-hole physics have concentrated on this inequality, see e.g. [4][5][6][7][8][9][10]. Because entropy and shear viscosity are common thermophysical properties with a clear definition for ordinary matter, this inequality has been tested with experimental data for ordinary fluids like the rare gases, H 2 , N 2 , CH 4 , and CF 4 [11]. For all systems studied so far it has been shown that κ>κ 0 holds well. Interestingly, the minimum value κ min of κwas observed in the vicinity of the critical point of ordinary systems [11], of high-energy matter [12] and also of high-temperature superconductors [13]. It was observed for ordinary fluids [11] that the minimum value κ min of κof the rare gases and other small molecules is in the range between κ min ≈(9 K 100) κ 0 . Roughly speaking κ min is in the same order of magnitude as κ 0 . This result is not necessarily expected for these fluids because the lower bound κ 0 is obtained from theoretical considerations of very high-energy physical phenomena including black holes using string theory methods [1].
There are much more thermophysical properties than η and s. The question is whether there are more combinations of other thermophysical properties which are bound from below. This question will be addressed in a heuristic manner. Existing inequalities are summarized in table 1. We show that their accepted lower bounds can be recovered with our conjecture. New inequalities are presented in sections 3.2 and 3.3, whereas in sections 3.1 and 3.4 special conclusions are drawn for properties, which have the dimension time times temperature.

Uncertainty relations
We start with the Heisenberg uncertainty principle (HUP) originally formulated as p 1 x 1 ∼h [14]. Nowadays the HUP is almost exclusively presented as Here p and x are the momentum and position, respectively, and for any observable A it can also be related to the uncertainty of measurement [15]. The Heisenberg uncertainty principle sets a rigorous and very fundamental limit on the product ΔpΔx. It is one of the cornerstones of quantum mechanics.
During the last years many other uncertainty relations of the type ΔAΔBC have appeared in the literature [14,, see table 1. In principle A and B are experimentally accessible properties which form a complementary pair. Examples for A, B, and C are energy E and time t with  = C 2, position x and time t with  = C G c 4 , but also temperature T and length x with  = C c k B . It might be interesting to note that also thermodynamic properties like temperature T, Helmholtz energy F, chemical potential μ, volume V, and pressure Π do occur in these formulations. Indeed, the role of thermodynamic uncertainty relations as fundamental bounds in biological and chemical physics are currently under investigation [35][36][37]. It turns out that the bound C>0 can always be expressed in terms of the fundamental physical constants. It should be stressed that the inequalities given in table 1 are obtained from sophisticated gedanken experiments as well as elaborate and sometimes lengthy treatments of the underlying statistical, quantum, thermodynamic, or gravitational theories.

Generalized uncertainty principle
We are now looking for a simple merely heuristic way to obtain the lower bounds of the uncertainty relations presented in table 1. By definition both ΔA>0 and ΔB>0 are positive quantities. The question is whether or not they might become arbitrarily small. If not then they are restricted from below as ΔAA min >0 and ΔBB min >0. A min and B min are the smallest values possible for A and B. This is a very fundamental restriction [41] which, however, seems to arise naturally in the framework of quantum gravity. This theory should merge general relativity and quantum theory and can be traced back to some remarks given by Einstein [42]. However, the theory itself is far from being complete nor is there any consensus of how it should look like. But it was shown in many different ways from first principles that merging quantum theory and general relativity will lead to a minimal length x 0 >0, see e.g. [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57]. The consequence is that a distance x with x<x 0 does not have any meaning and therefore is not accessible. As a result we always have xx 0 >0.

The Planck units
It is often assumed but not proved that the lower bounds x 0 and Δx 0 are given by the Planck length ℓ P . We therefore set D = = ℓ x x 0 0 P . The Planck units were originally introduced by Max Planck in 1900 [70] in order to create an unbiased system of units of the fundamental physical dimensions time, mass, and length. In his opinion this system should be unique and valid throughout the whole universe. The Planck units can be obtained by a dimensionful correct combination of the speed of light in vacuum c, the Planck constant h, the Boltzmann constant k B , and the Newtonian constant of gravitation G. In recent years the originally proposed units have been augmented by temperature, density, and other properties [71][72][73][74][75]. A small collection of these units is given in  [71] or as F P =F max /4=c 4 /(4G) [41,76]. In the following we use the Planck units of time and length as a natural lower bound, the Planck units of density and temperature as a natural upper bound for these properties. According to Gibson [71] the Planck unit of any product C=AB might be obtained by a dimensional correct combination of the corresponding Planck units of A and B. Combinations with ℓ P , t P , T P , and  P are considered to be a natural bound to any property examined in this work. The lower limit of the HUP will then result in . We now apply this simple combination scheme to all uncertainty relations listed in table 1. In addition to table 2 the Planck units of entropy = S k P B and linear momentum = p m c P P are also used [71,75]. In the case of the indeterminacy relations (3)-(5), (7), (8), (10), (11), (13), and (14) the exact lower bounds are recovered. For the remaining five relations only a factor of (1/2) or (1/4π) is missing. Obviously our heuristic scheme works quite well and the simple use of Planck units enables us to recover the lower bounds of the indeterminacy relations quite easily.

Viscosity and entropy-the Kovtun-Son-Starinets conjecture
We start with the Kovtun-Son-Starinets conjecture already mentioned in the introduction. In (1) we have seen, that the fundamental lower bound exists. The lower bound of this inequality can be recovered easily with our afore mentioned treatment with the Planck units. We write the corresponding Planck ratio κ P as Here  n = ( ) G c P 1 2 is the Planck unit of the kinematic viscosity [71] and = ℓ V P P 3 the Planck volume. The result is κ 0 =κ P /(4π). Beside the factor 1/(4π) the lower bound of the KSS conjecture κ 0 is recovered with this simple procedure. This is in full accord with the observations summarized in table 1. Beside a numerical factor which is close to one the lower bounds of the corresponding inequalities can be obtained from the Planck units.  [70][71][72][73][74][75]. The numerical values are based on the CODATA recommended values of the fundamental physical constants [77]. α is the fine structure constant. Note that some authors suggest to use corrected Planck units which are deduced by substituting 4G for G [41].

Constant Definition Value
Planck length ℓ P  ( ) G c 3

Thermal conductivity, diffusion, and entropy
Shear viscosity η and density of entropy s=S/V are transport and equilibrium properties of fundamental importance in e.g. physical chemistry and chemical engineering. The question is if there are other thermophysical properties and combinations thereof which obey an inequality like (1). We consider the two transport coefficients thermal conductivity l T and self diffusion D 11 in relation to the density of entropy s. It was discussed by Rosenfeld [78,79] and others [80][81][82] that a possible relation between s and the transport coefficients η, λ T and D 11 , respectively, might exist. Therefore, its reasonable to study the following three ratios. According to our proposal we obtain the lower bounds from the corresponding Planck units. In the case of the ratio l s T thermophysical data can be found in [83] for the same gases as in the case of η/s [11]. The results are shown in figure 1. In all cases an experimentally obtained minimum value of l s T can be found in the vicinity of the critical temperature. However, the temperature dependence of the critical isobars is more complicated compared to the behaviour of κ. Nevertheless, the minimum values can be found in the range of l » ¼ -( ) ( )· s 2 10 10 m s T min 8 2 which is twenty orders of magnitude above the proposed theoretical limit l s TP P . Experimentally determined self-diffusion coefficients D 11 of pure substances over wide ranges of temperature and pressure are much harder to find [84]. Therefore, we concentrate ourselves to a region in the vicinity of the critical point where some experimental data can be found [84][85][86][87][88]. We just mention the result for methane given by Oosting and Trappeniers [86]. They report a value in the range of D 11 ≈(58.5 K 67)·10 −9 m 2 s −1 in the critical region. This gives an experimental value of » ¼ ( ) D s 4.2 4.8 11 · - D s 10 K s m kg 14 3 11P P . This is even 94 orders of magnitude above the Planck limit given in (5). Although the limits proposed in (4) and (5) strictly hold, at a first glance these two inequalities might be regarded as irrelevant. This is in strong contrast to the limit κκ 0 proposed in (1) which seems to be of relevance even in typical thermophysical applications.
Often, diffusion coefficients are tabulated as ñD 11 , ñbeing the mass-density of the system [89]. This leads to the inequality (6). Note that the lower bound  D s P P 11 P is given by the same Planck limit  k B as in the case of η/s. We use experimentally determined self-diffusion coefficients D 11 [85][86][87][88]90] as well as tabulated densities and entropies [83] at the critical point and obtain ñD 11 /s in the sequence neon, argon, krypton, xenon, hydrogen, and methane as 2.0·10 −12 , 1.1·10 −11 , 1.64·10 −11 , 2.5·10 −11 , 2.54·10 −12 , and 7.94·10 −12  (all in Ks), respectively. We notice that neon and hydrogen do not obey the inequality (6). In order to settle this shortcoming we might suppose that a factor of 1/(4π) is missing on its right-hand-side. This fact was already noticed for κwhere we found κ 0 =κ P /(4π). The missing term 1/(4π) can also be deduced from the temperature-length inequality given in table 1. Indeed,  p ( ) k 4 B is a lower bound to the afore mentioned experimental results for ñD 11 /s. We conclude that beside inequality (1) also inequality (6) seems to be of relevance for ordinary fluids. This, however, is a somewhat expected result provided that diffusion coefficients and viscosities show no anomalies at T C [91]. Indeed we notice that the experimentally determined ratio ( ) D s m 11 T C correlates very well with (η/s) min for these fluids at the critical temperature T≈T C [11]. In particular we obtain (η/s) min /( ) D s m 11 T C in the range between 2.0 and 2.15 for argon, krypton, xenon and hydrogen, 5.1 in the case of neon and 1.55 for methane. These ratios at least have the same order of magnitude for the six gases and (η/s) min approximately scales like ( ) D s m 11

The Wiedemann-Franz law
The Wiedemann-Franz law states that for metals the ratio of the thermal and electrical conductivity, λ T /σ, is directly proportional to the temperature T, see e.g. [92,93].
The proportionality constant L is known as the Lorenz number. It is approximately constant for a variety of metals and alloys. Beside various erroneous attempts L was first calculated correctly by Sommerfeld [94] using Fermi-Dirac statistics: In finite temperature experiments L≈L 0 is observed. Relation (7) does not only hold to a very good approximation for ordinary metals, alloys and degenerate semiconductors [92,93,95], but also for some Fermi liquids [96] in general. It has been shown by a number of theoretical and experimental observations that (8) is strictly valid for metals and alloys [97][98][99][100] as well as Fermi liquids including heavy fermionic systems like CeAl , CeCu , UPt 3 6 3 [101][102][103]. It also holds strictly for some quantum critical metals [96] and some non-Fermi liquids [104]. However it complety fails for superconductors due to Cooper-pairing of the electrons [102,105]. As outlined before we construct a Lorenz number L P by using the Planck units for thermal and electrical conductivity and the temperature. We set L P as a lower limit to L and restrict ourselves to electrons in metals which behave like a Fermi liquid and obtain First we note that  L L 0 P holds. This means that our proposed lower limit L P is in good accordance with the rigorously calculated Sommerfeld limit L 0 . Second, experimental results at various temperatures show that the prediction LL P is fulfilled for a large number of metals, alloys and degenerate semiconductors [93,95]. In contrast to the experimental results compiled in [93,95] in the case of silver Gloos et al [106] observed a severe deviation from the Wiedemann-Franz law between temperatures of 2K and 9K. In particular they measured L exp =(0.1 K 2)·10 −8 (V/K) 2 <L 0 . According to Gloos et al [106] the unexpected wide range of experimental results is based on the different purities and treatments of the Ag samples. Nevertheless, even in this exceptional case L exp >L P is observed. Hence we can conclude, that the Planck-limit L P of the Lorenz number holds for fermionic systems.

Time and temperature
By using some heuristic arguments de Sabbata and Sivaram [23,24] obtained the indeterminacy relation 12 Ks by introducing torsion in general relativity. Assuming again that the uncertainties ΔT and Δt are limited by the corresponding Planck units T P and t P , we obtain exactly the same bound B as was given in [23,24]. This relation was tested in a typical laboratory experiment by Gillies and Allison [29,30]. These authors analyze the laser-induced fluorescence decay of nanoparticles of YAG:Ce. They obtained an experimentally determined minimal value in the range of 12 Ks in full accord with the lower bound proposed in [23,24]. Interestingly (ΔTΔt) min is only (6...262) times above the theoretical limit (ΔTΔt) 0 , which is obtained from theoretical considerations in the field of general relativity. Again it should be stressed that the lower bound which is approached by experiment has the dimension temperature times time, exactly the same as for η/s and ñD 11 /s. A quite similar relation was developed by Hod [107] from quantum information theory and thermodynamics. He obtained the inequality τ is the relaxation time of a perturbed thermodynamic system. This bound is called 'TTT' (time times temperature) [108] and it may be used as a quantitative way to explain the third law of thermodynamics [107]. A similar inequality was already conjectured by Sachdev [109] and the bound on the relaxation time is applied to e.g. quantum critical phenomena [110] as well as in the framework of incoherent metallic transport [111].

Photon and graviton lifetimes
In this last example we dare to address two fundamental questions in physics. What is the mass of the photon and the graviton and do they have an infinite lifetime? We propose that the uncertainty relation   D D M t c 2 given by Landau and Peierls [16] and Aharonov and Reznik [28] might be of relevance in view of the mass and lifetime limits, ΔM and Δt, of the photon [112,113] and graviton [112,114]. First we mention that the same lower limit can be obtained from the product of the Plack units which also gives  D D M t m t P P =  » c 2 -· 1.17 10 51 kgs. We test this inequality bearing in mind that the lower limits of the mass ΔM [112] and especially of the lifetime Δt [113,114] of the photon and graviton are still to some extend speculative and may both vary by several orders of magnitude. By using the lowest limits given in the literature we obtain for the photon ΔMΔt≈3.15·10 −37 kgs and in the case of the graviton ΔMΔt≈1.00·10 −38 kgs. We see that the proposed bound on   D D M t c 2 holds very well.

Discussion and conclusion
We have seen in table 1 that our simple heuristic procedure recovers the limits of the indeterminacy relations given in the literature. However, sometimes a factor of 1/2 or 1/(4π) is missing. This might be a drawback. But e.g. in the case of the uncertainty relation   b D D p t c even in extensive calculations the rhs is given with β=1 [16] or β=1/2 [34]. We have formulated new bounds for the ratio of important thermophysical properties, namely , respectively. In the case of the first two inequalities experimental results are several orders of magnitude larger than the bounds given on the rhs Experimental results for ñD 11 /s are close to or even below the proposed limit of  k B . However, the inequality holds if a factor of 1/(4π) is introduced in the rhs which then reads  p ( ) k 4 B . The question in general is if this bound is of similar importance as the KSS-bound discussed in section 3.1. We also have tackled the Wiedemann-Franz law. In the case of fermionic systems we have proposed the lower limit of 2 . This inequality holds well, the theoretical result of Sommerfeld as well as experiments are roughly 200 times above this limit. The lower bounds of the inequalities contain the fundamental physical constants ÿ, k B , e, c, and G. In the course of these studies we notice that the bounds are approached quite closely by ordinary (not exotic high energy) matter if only the first three of these fundamental constants occur in the corresponding Planck limit. We have seen this in the cases of h s,  D s 11 , l s ( ) T T , and ΔtΔT, but not for D 11 /s and l s T where the Newtonian constant G and the speed of light in vacuum c enter the corresponding lower bounds. G is the characteristic fundamental constant in general relativity and c in special relativity. On account of these observations we can conclude that effets from general relativity do not play a significant role for ordinary bulk matter behaviour. If in contrast to this the lower bound of an inequality is given by the Planck limit  k B then this limit might be of relevance in every day physico-chemical investigations. This consequence should be tested for a variety of other physico-chemical properties.

Conflict of interest
The authors declare that they have no conflict of interest.