Query on Negative Temperature, Internal Interactions and Decrease of
  Entropy
Microsoft Word - negEntr.doc
Query on Negative Temperature, Internal 
Interactions and Decrease of Entropy  
Yi-Fang Chang 
Department of Physics, Yunnan University, Kunming, 650091, China 
(e-mail: yifangchang1030@hotmail.com) 
Abstract:  After negative temperature is restated, we find that it will derive 
necessarily decrease of entropy. Negative temperature is based on the Kelvin scale 
and the condition dU>0 and dS<0. Conversely, there is also negative temperature for 
dU<0 and dS>0. But, negative temperature is contradiction with usual meaning of 
temperature and with some basic concepts of physics and mathematics. It is a question 
in nonequilibrium thermodynamics. We proposed a possibility of decrease of entropy 
due to fluctuation magnified and internal interactions in some isolated systems. From 
this we discuss some possible examples and theories. 
Keywords: entropy; negative temperature; nonequilibrium. 
PACS: 05.70.-a, 05.20.-y 
1.Restatement of Negative Temperature 
In thermodynamics negative temperature is a well-known idea proposed and 
expounded by Ramsey [1] and Landau [2], et al. 
Ramsey discussed the thermodynamics and statistical mechanics of negative 
absolute temperatures in a detailed and fundamental manner [1]. He proved that if 
the entropy of a thermodynamic system is not a monotonically increasing function of 
its internal energy, it possesses a negative temperature whenever XUS )/(
negative. Negative temperatures are hotter than positive temperature.  He pointed out: 
from a thermodynamic point of view the only requirement for the existence of a 
negative temperature is that the entropy S should not be restricted to a monotonically 
increasing function of the internal energy U. In the thermodynamic equation relating 
TdS and dU, a temperature is 
)/( XUST .                                            (1) 
An assumption, entropy S increases monotonically with U, is not necessary in the 
derivation of many thermodynamic theorems. Negative temperatures are hotter than 
infinite temperature. Negative temperature is an unfortunate and misleading. If the 
temperature function had been chosen as 1/T, then the coldest temperature would 
correspond to -
for this function, infinite temperatures on the conventional scale 
would correspond to 0, and negative temperatures on the conventional scale would 
correspond to positive values of this function. 
Ramsey proposed: One of the standard formulations of the second law of 
thermodynamics must be altered to the following: It is impossible to construct an 
engine that will operate in a closed cycle and prove no effect other than (a) the 
extraction of heat from a positive-temperature reservoir with the performance of an 
equivalent amount of work or (b) the rejection of heat into a negative-temperature 
reservoir with the corresponding work being done on the engine. A thermodynamic 
system that is in internal thermodynamic equilibrium, that is otherwise essentially 
isolated.  Negative temperatures are applied to some mutually interacting nuclear spin 
system. Klein justified Ramsey s criteria for systems capable of negative absolute 
temperatures [3]. Only premise of Ramsey s statement is Kelvin definition (1). Even 
he implied that entropy might decrease with U, i.e., in an original definition of 
temperature     
T=dU/dS,                                                  (2) 
when dU>0 and dS<0, T<0. 
Intuitively, the physical meaning of temperature is that it describes whether a 
body is hot or cold
[4]. A definition of an absolute thermodynamic temperature 
scale is proportional to the quantity of heat. Maxwell s definition is that the 
temperature of a body is its thermal state considered with reference to its power of 
communicating heat to other bodies, which was adopted substantially unchanged by 
Planck and Poincare [5]. In microscopic thermodynamics temperatures are related 
with the states of molecular motions. Kelvin temperature scale is defined by the 
relationship (1). 
It is a little different that Landau proved negative temperature. In Landau s book 
<Statistical Physics> [2] negative temperature was stated as following: Let us 
consider some peculiar effects related to the properties of paramagnetic dielectrics. 
Here the interaction of these moments brings about a new magnetic spectrum, which 
is superposed on the ordinary spectrum. From this the entropy is     
ln nnmag EE
gNS ,                             (3) 
where N is the number of atoms, g is the number of possible orientations of an 
individual moment relative to the lattice, nE are the energy levels of the system of 
interacting moments, and nE is the average as the ordinary arithmetic mean. This 
shall regard the atomic magnetic moments fixed at the lattice sites and interacting 
with one another as a single isolated system. Further, there has the interesting result 
that the system of interacting moments may have either a positive or a negative 
temperature. At T=0, the system is in its lower quantum state, and its entropy is 
zero  [2]. In fact, the temperature T=0 (absolute zero) is impossibly achieved. As the 
temperature increases, the energy and entropy of the system increase monotonically. 
At T= , the energy is nE and the entropy reaches its maximum value Nlng; these 
values correspond to a distribution with equal probability over all quantum states of 
the system, which is the limit of the Gibbs distribution as T .
The statement of 
original negative temperature is based on the two premises: entropy of the system 
increase monotonically, and the Gibbs distribution holds. From this some strange 
arguments [2] are obtained: 
(a). The temperature T=-
is physically identical with T= ; the two values 
give the same distribution and the same values of the thermodynamic quantities for 
the system.
According to the general definition, temperature cannot be infinite, since 
the quantity of heat or molecular motion all cannot be infinite. Negative temperature, 
even negative infinite temperature is stranger. In the same book <Statistical Physics>, 
Landau proved a very important result that the temperature must be positive: T>0 [2]. 
Moreover, T= =-
is excluded by mathematics. 
(b). A further increase in the energy of the system corresponds to an increase in 
the temperature from T= , and the entropy decreases monotonically.
(c). At T=0- the energy reaches its greatest value and the entropy returns to zero, 
the system then being in its highest quantum state.
This obeys Nernst s theorem, but 
in which the quantity of heat is zero at T=0, while at T=0- it possesses highest 
quantum state! I do not know whether T=0=T=0- holds or not. 
(d). The region of negative temperature lies not below absolute zero but above 
infinity , i.e., negative temperatures are higher than positive ones . 
2.Query on Negative Temperature  and Decrease of Entropy 
In a previous paper [6], we proved that since fluctuations can be magnified due 
to internal interactions under a certain condition, the equal-probability does not hold. 
The entropy would be defined as  
rr tPtPktS )(ln)()( .                                      (4) 
From this or lnkS in an internal condensed process, possible decrease of 
entropy is calculated. If various internal complex mechanism and interactions cannot 
be neglected, a state with smaller entropy (for example, self-organized structures) will 
be able to appear. In these cases, the statistics and the second law of thermodynamics 
should be different, in particular, for nonequilibrium thermodynamics [7,8]. Because 
internal interactions bring about inapplicability of the statistical independence, 
decrease of entropy due to internal interactions in isolated system is caused possibly. 
This possibility is researched for attractive process, internal energy, system entropy 
and nonlinear interactions, etc [6]. 
In fact, negative temperature derives necessarily decrease of entropy. We think, 
T= =-
is only a finite threshold temperature cT , which corresponds to a value 
of entropy from increase to decrease, and this entropy is a maximum Nlng. According 
to the basic equation of thermodynamics, i.e., Euler equation [9], 
S .                                            (5) 
If energy is invariance, corresponding temperature should be 
.                                 (6) 
This value should be testable and measurable. Such temperature and energy increase 
continuously, and entropy decreases to a minimum, but cannot be zero. Of course, 
energy passes necessarily from negative temperature system
to positive temperature 
system. 
Next, the Gibbs distribution is     
nAew / .                                              (7) 
This finds the probability nw of a state of the whole system such that the body 
concerned is in some definite quantum state (with energy nE ), i.e., a microscopically 
defined state , and is suitable that the system is assumed to be in equilibrium
[2]. 
So long as assume that the Gibbs distribution hole always, it is necessary for negative 
temperature. But, in the above example and laser that is another example applied 
negative temperature, these states are already unstable or metastable nonequilibrium 
states with higher energy. Bodies of negative temperature are also completely unstable 
and cannot exist in Nature [2]. The entropy of a body is a function only of its internal 
energy [2]. In states with negative temperature, the crystal be magnetized in a strong 
magnetic field, then the direction of the field is reversed so quickly that the spins 
cannot follow it [2]. This system is in a nonequilibrium state, and its internal energy 
and entropy are different. Laser should be an ordering process with decrease of 
entropy. 
Generally, the Gibbs distribution for a variable number of particles is [2]     
nNAew /)( ,                                        (8) 
where 
is the thermodynamic potential. From this the distributions are different for 
the number N of particles. The number N should be different in magnetic field with 
reversed direction. 
The above statement of negative temperature  proves just that entropy is able to 
decrease with internal interactions in an isolated system. The experimental study 
requires that the spin system be well isolated from the lattice system [1]. This 
isolation is possible if the ratio of spin-lattice to spin-spin relaxation times is large [2].  
This may describes the Figure 1, which is namely Fig.1 [1] and Fig.10 [2], in which 
the finite threshold value cT corresponds to only a maximum point dS/dE=0. 
Fig.1  negative temperature
According to Eq.(1) or Eq.(2), when the condition dU>0 and dS<0 hold, negative 
temperature will be obtained. Conversely, if dU<0 and dS>0, we will also derive 
negative temperature. But, Eq.(2) originates from Clausius entropy dS=dQ/T, so 
whether above conditions could hold or not? According to Eq.(5), if 
ii NYX =0, 
T=U/S. In Fig.1, since U>0 (E>0) and S>0, then T>0. From this case, we cannot 
obtain negative temperature, and gNET cc ln/ . 
In another book <Principles of General Thermodynamics> [5], Kelvin 
temperature (1) of thermodynamic system may be either positive or negative, 
according to whether, as the system passes through stable states with fixed parameters, 
the entropy increases or decreases with increasing energy. This is different with 
Landau s statement. In fact, this statement seems to imply that the negative 
temperature is unnecessary, so long as the entropy decreases with increasing energy.  
A normal system can assume only positive Kelvin temperatures. ± A system at a 
negative Kelvin temperature is in a special state. For if this were not true we could, by 
definition, do work on the system adiabatically and prove that the system was at a 
positive Kelvin temperature. A system is capable of attaining negative Kelvin 
temperatures if for some of its stable states the entropy decreases for increasing 
energy at fixed values of the parameters.
Further, as examples, the entropy for a monatomic gas is given by   
'lnln)2/3( SRTRS .                                  (9) 
Based on this, T and the density 
cannot be negative. The systems are nuclear spin 
ones in a pure lithium fluoride (LiF) crystal spin lattice relaxation times were as large 
as 5 minutes at room temperature while the spin-spin relaxation time was less than 
seconds. The systems lose internal energy as they gain entropy, and the reversed 
deflection corresponds to induce radiation. The sudden reversal of the magnetic field 
produces negative temperature for the Boltzmann distribution [10]. Next, this book [5] 
discussed heat flow between two systems A and B at unequal temperatures, and 
derives  
AdQ .                                        (10) 
Here let a heat quantity AdQ flow into A from B, so there should be 0AB TT , 
which is also consistent with inequality (10). Further, according to an efficiency of 
heat engines 
.                                              (11) 
Here if either temperature is negative, the efficiency will be greater than unity. 
Assume that be possible, this will be testified and infinite meritorious and beneficent 
deeds. The results arrived at for negative temperatures which are strange to our 
intuition have no practical significance in the field of energy production.
But, 
°syst e m at negative Kelvin temperatures obey the second law and its many 
corollaries. 
Of course, it would be useless to consume work in order to produce a 
reservoir at a negative temperature which can be used to operate a very efficient heat 
engine
[5]. Therefore, this seems to imply that negative temperature is introduced 
only in order to obey the second law of thermodynamics.     
There is the same efficiency of a Carnot engine applied by Ramsey [1]: 
.                                       (12) 
Here various results of existence of the machine were discussed in order to be not in 
contradiction to the principle of increasing entropy. 
For an equal-temperature process, there is a simple result:         
dS=(dU+PdV)/T,                                          (13) 
where U is the internal energy of system. A general case is (dU+PdV)>0, dS>0 for 
usual temperature T>0; dS<0 if T<0. Further, if T>0 and (dU+PdV)<0, for example, a 
contractive process is dV<0, dS<0 is possible [6]. 
In fact, so long as dS<0, the negative-temperature is unnecessary. Otherwise, 
one of the standard formulations of the second law of thermodynamics must be 
altered to the following: It is impossible to construct an engine that will operate in a 
closed cycle and prove no effect other than (1) the extraction of heat from a 
positive-temperature reservoir with the performance of an equivalent amount of work 
or (2) the rejection of heat into a negative-temperature reservoir with the 
corresponding work being done on the engine.
The experimental study requires that 
the spin system be well isolated from the lattice system. This isolation is possible if 
the ratio of spin-lattice to spin-spin relaxation times is large [1]. 
The condition in which there are more atomic systems in the upper of two energy 
levels than in the lower, so stimulated emission will predominate over stimulated 
absorption. This condition may be described as a negative temperature. 
In a word, negative temperature is a remarkable question, in particular, for 
nonequilibrium thermodynamics. Is it a fallacy? From Kelvin scale one obtained 
infinite temperature and negative temperature, which is inconsistent with other 
definition of temperature, and with some basic concepts of physics and mathematics. 
Moreover, negative temperature
is confused easily with an absolute zero defined 
usually by negative 273.16C. 
3. Some Possible Examples for Decrease of Entropy 
For a mixture of the ideal gases, the increased entropy is          
jj xRndS
ln .                                        (14) 
If the interactions of two mixed gases cannot be neglected, the change of the free 
energy will be [9]: 
7         
212211 )lnln( xxxxxxRTGG if ,                         (15) 
where )/( 2111 nnnx , etc. Then the change of entropy of mixing will be         
]/)([)lnln(]/)([ 212211 TxxxxxxRTdGdS .         (16) 
When >0, i.e., the interaction is an attractive force, there is probably dS<0. For 
instance, for 21 xx 1/2,          
dS Rln2 )4/( T .                                     (17) 
When >(4Rln2)T, dS<0 is possible [8]. 
Many protons mix with electrons to form hydrogen atoms, a pair of positive and 
negative ions forms an atom, and various neutralization reactions between acids and 
alkalis form different salts. These far-equilibrium nonlinear processes form some new 
self-organize structures due to electromagnetic interactions. They should be able to 
test increase or decrease of entropy in isolated systems. 
The total rate of production of entropy is [11]:          
total
.                                   (18) 
If the heat current QJ >0, the total rate will be dS/dt>0 for 21 TT , and dS/dt<0 for 
21 TT . We discussed an attractive process based on a potential energy         
U i ,                                                 (19) 
in which entropy decreases [6]. 
Using a similar method of the theory of dissipative structure in non-equilibrium 
thermodynamics, we derived a generalized formula, in which entropy may increase or 
decrease, the total entropy in an isolated system is [6]:         
ia dSdSdS ,                                            (20) 
in which adS is an additive part of entropy, and idS is an interacting part of 
entropy. Further, the theory may be developed like the theory of dissipative structure. 
Barbera discussed the principle of minimal entropy production, whose field equations 
do not agree with the equations of balance of mass, momentum and energy in two 
particular cases. The processes considered are: heat conduction in a fluid at rest, and 
shear flow and heat conduction in an incompressible fluid [13]. 
Eq.(4) is similar to a generalization of the Boltzmann-Gibbs entropy functional 
proposed by Tsallis [14], which given by a formula:        
iq PPkS ln .                                                 (21) 
where iP is the probability of the ith microstate, the parameter q is any real number,       
)0(),1()1(ln 11 ffqf qq                                (22) 
When q 1, it reduces to 
      
ii PPkS ln .                                            ( 23) 
The entropy of the composite system BA
verifies 
)()()1()()()( BSASqBSASBAS qqqqq .                    (24) 
Our conclusions are consistent quantitatively with the system theory [6], in 
which there is [15]         
)()()( 21 SSS .                                       (25) 
This corresponds to Tsallis entropy [14]:         
q ,                                           (26) 
which is nonextensive for Eq.(26) when q>1. Both seem to exhibit decrease of 
entropy with some internal interactions. 
For more general case, statistical independence in mathematics corresponds to 
the independence of probability, i.e., addition of probabilities is 
ipP . But, 
addition of dependent probabilities is 
.)....()1(....)(
)()()....(
AAApAAAp
AApApAAAP
                
(27) 
Therefore, from independence to interrelation, probability decrease necessarily, whose 
amount is determined by interaction strength. Correspondingly, entropy on mixture of 
different systems should decrease. In fact, any internal interaction in a system 
increases already relativity and orderliness. 
In present theory, the superfluid helium and its fountain effect must suppose that 
the helium does not carry entropy, so that the second law of thermodynamics is not 
violated [9]. It shows that the superfluids possess zero-entropy, but it cannot hold 
because zero-entropy corresponds to absolute zero according to the third law of 
thermodynamics. For the liquid or solid 3He the entropy difference [9] is 
sl SSS >0 (for higher temperature), =0 (for T=0.3K), <0 (for lower temperature). 
Such a solid state with higher entropy should be disorder than a liquid state in lower 
temperature! 
Otherwise, in chemical thermodynamics, the entropy of formation is a variant 
with pressure. In general, any chemical reaction can take place in either direction. 
In a word, according to the second law of thermodynamics, all systems in Nature 
will tend to heat death
[16], while product will be impossible. But, world is not 
pessimistic always. The gravitational interactions produce various ordered stable stars 
and celestial bodies. The electromagnetic interactions produce various crystals and 
atoms. The stable atoms are determined by electromagnetic interaction and quantum 
mechanics. According to the second law of thermodynamics, they should be unstable. 
The strong interactions produce various stable nuclei and particles. A free proton is 
stable, which testifies that some quarks may form a structure by strong interaction. 
Theses stabilities depend mainly on various internal interactions and 
self-organizations. These attractive interactions correspond to decrease of entropy in 
our theory [6]. Increase of entropy corresponds to repulsive electromagnetic 
interactions with the same changes and weak interactions. Any stable objects and their 
formations from particles to stars are accompanied with internal interactions inside 
these objects, which have implied a possibility of decrease of entropy. The stability in 
Nature waits our study and research. 
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