below absolute zero?
Jack Sarfatti
This is from a thread in rec.arts.startrek.tech which you physics junkies
should start looking at if you want to generate students and earn your
academic salaries in a time of decreasing educational investment - which may
or may not change under Clinton:
I'm not sure of what was originally intended. I just caught the posts
whipping by at warp 10 and did not really digest it. There is in quantum
physics such a thing as a negative spin temperature - this would be formally
below absolute zero - but that is misleading since any negative temperature
is "hotter" than any positive temperature. Indeed -.0000000000.....1 degrees
Kelvin is hotter than infinite positive temperature. This is because of the
way entropy varies with respect to energy in quantum systems with discrete
energy levels. A negative temperature means that more particles sit in the
upper energy level than in the lower energy level for the simple case of
a two level system like a spinning electron, proton or even neutron, or atom
of spin 1/2 in a magnetic field.
Thought problem for Star Fleet Academy Cadets from Admiral Sarfatti:
How does a reversible Carnot heat engine behave if the hot reservoir is
at a negative quantum temperature while the cold reservoir is at a positive
classical temperature? What kind of Star Fleet devices can you make with
this idea? Hint: use the second law of thermodynamics.
Jon Noring
>I'm not sure of what was originally intended. I just caught the posts
>whipping by at warp 10 and did not really digest it. There is in quantum
>physics such a thing as a negative spin temperature - this would be formally
>below absolute zero - but that is misleading since any negative temperature
>is "hotter" than any positive temperature. Indeed -.0000000000.....1 degrees
>Kelvin is hotter than infinite positive temperature. This is because of the
>way entropy varies with respect to energy in quantum systems with discrete
>energy levels. A negative temperature means that more particles sit in the
>upper energy level than in the lower energy level for the simple case of
>a two level system like a spinning electron, proton or even neutron, or atom
>of spin 1/2 in a magnetic field.
>
>Thought problem for Star Fleet Academy Cadets from Admiral Sarfatti:
>How does a reversible Carnot heat engine behave if the hot reservoir is
>at a negative quantum temperature while the cold reservoir is at a positive
>classical temperature? What kind of Star Fleet devices can you make with
>this idea? Hint: use the second law of thermodynamics.
If you'd like a clue, consult chapter 14 in the book "Heat and
Thermodynamics", by Mark W. Zemansky, McGraw Hill, 1968).
Jon Noring
--
Charter Member of the INFJ Club.
Now, if you're just dying to know what INFJ stands for, be brave, e-mail me,
and I'll send you some information. It WILL be worth the inquiry, I think.
=============================================================================
| Jon Noring | nor...@netcom.com | I VOTED FOR PEROT IN '92 |
| JKN International | IP : 192.100.81.100 | Support UNITED WE STAND! |
| 1312 Carlton Place | Phone : (510) 294-8153 | "The dogs bark, but the |
| Livermore, CA 94550 | V-Mail: (510) 417-4101 | caravan moves on." |
=============================================================================
Who are you? Read alt.psychology.personality! That's where the action is.
john baez
>Thought problem for Star Fleet Academy Cadets from Admiral Sarfatti:
>How does a reversible Carnot heat engine behave if the hot reservoir is
>at a negative quantum temperature while the cold reservoir is at a positive
>classical temperature? What kind of Star Fleet devices can you make with
>this idea? Hint: use the second law of thermodynamics.
I won't answer the above question, but I will give you space cadets out
there a hint. When you put a body with negative temperature in contact
with one of positive temperature, the temperature of the "cold" one *rises*
while the temperature of the "hot" one *falls*. This is easy to see (for
physicists) if you recall that 1/kT is beta, which is dS/dE. So (forgetting
Boltzmann's constant k), dS/dE = T. Thus a system with negative temperature
gets more entropy if you *decrease* its energy, while a system with positive
temperature gets more entropy if you *increase* its energy. Systems like
to increase their entropy, so energy will like to flow from the negative
T system to the positive T system.
This is another reason why it sort of makes sense to think of negative
temperatures as being temperatures that are greater than infinity.
I suppose readers who don't already trust me will think I'm nuts, but
this is really all pretty well known. I think Scientific American had
a nice article on systems with negative temperature a while back.
Michael Jay Malak
>An easy way to construct "lower than zero"-temperatures:
>All you need is a magnetic material and a strong magnet.
>Then 1) Cool the whole thing and switch an the magnet.
>2) Wait until ALL spins are orientated upwards.
>3) Change the direction of the field.
>So the spin are all looking against the lines of the field.
>The is identified with negative absolute temperatures.
>But this is only a statistical definition.
Well, let's be more precise here. The statistical-mechanics
definition of absolute temperature is that 1/T = dS/dE (that is, the
partial derivative of entropy with respect to energy). All we care
about is the sign. For most physical systems, the energy can be
arbitrarily large, and the entropy always increases with energy.
But let us consider a system which has a maximum possible energy.
A simple one is the case of two spins in a magnetic field. A particle
can have one of two possible energies, represented here as ``+'' and ``-''.
A particle in the `+' state has an energy E, and one in the `-' state
has energy 0. The possible states of the whole system are:
State Energy Entropy
-- 0 0
+- OR -+ E S = k ln 2
++ 2E 0
The middle (energy E) system has two indistinguishable possible
configurations, so its entropy is nonzero. Increasing the energy
of the system to 2E _reduces_ the entropy.
It doesn't make much sense to speak of a `temperature' for such a small
system, but you can consider a system of a large number of spins:
as you add energy, the entropy increases until the total energy is 1/2
the maximum energy; after that, the entropy decreases. According to
the definition of absolute temperature, the system has a negative
temperature in this higher-energy regime.
Intuitively, you would think that adding energy to a system
should make it hotter. This is correct. Negative temperatures
are hotter than positive ones!
This isn't nearly a complete discussion; most textbooks on thermodynamics
or statistical mechanics have a few pages on negative temperatures.
--
Michael Malak | 1. All syllogisms have three parts.
mmalak@looking_glass.caltech.edu | 2. Therefore, this is not a syllogism.
David E. Brahm
spin system):
> Intuitively, you would think that adding energy to a system should make
> it hotter. This is correct.
I thought I'd point out (though it's tangential to the original discussion)
that gravitational systems do not obey your intuition. A gravitationally
bound collection of hydrogen, for example, radiates away energy, thus
collapsing and becoming hotter! (Eventually it becomes a star.) This
system has negative heat capacity. Since heat flow then makes hot things
hotter and cold things colder, for gravitational systems higher entropy is
associated with greater inhomogeneity. That's why the homogeneous early
universe is considered a very low-entropy state, and why the ultimate
result of graviational collapse, black holes, have entropy associated with
them.
I've just been reading this stuff in "The Physical Basis of the Direction
of Time", ch.5. Great book!
--
Staccato signals of constant information, | David Brahm, physicist
A loose affiliation of millionaires and | (br...@cco.caltech.edu)
billionaires and Baby ... |---- Carpe Post Meridiem! --
These are the days of miracle and wonder, | Disclaimer: I only speak
And don't cry, Baby, don't cry, don't cry. | for the sensible folks.
Markus Pristovsek
All you need is a magnetic material and a strong magnet.
Then 1) Cool the whole thing and switch an the magnet.
2) Wait until ALL spins are orientated upwards.
3) Change the direction of the field.
So the spin are all looking against the lines of the field. The is identified
with negative absolute tempratures. But this is only a statistical definition.
Todd Pedlar
>
>I won't answer the above question, but I will give you space cadets out
>there a hint. When you put a body with negative temperature in contact
>with one of positive temperature, the temperature of the "cold" one *rises*
>while the temperature of the "hot" one *falls*. This is easy to see (for
>physicists) if you recall that 1/kT is beta, which is dS/dE. So (forgetting
>Boltzmann's constant k), dS/dE = T. Thus a system with negative temperature
>gets more entropy if you *decrease* its energy, while a system with positive
>temperature gets more entropy if you *increase* its energy. Systems like
>to increase their entropy, so energy will like to flow from the negative
>T system to the positive T system.
>
>This is another reason why it sort of makes sense to think of negative
>temperatures as being temperatures that are greater than infinity.
>
>I suppose readers who don't already trust me will think I'm nuts, but
>this is really all pretty well known. I think Scientific American had
>a nice article on systems with negative temperature a while back.
True - also Kittel and Kroemer do a good job of explaining the notion
of negative temperatures in their text Thermal Physics.
tkp
______________________________________________________________________________
Todd K. Pedlar !
Graduate Student ! ...said one termite to the other at Smokey's
Department of Physics ! Bar & Grill, " Is the bartender here? "
Northwestern University !
------------------------------------------------------------------------------
atm...@vax.oxford.ac.uk
> In article <25...@galaxy.ucr.edu>, ba...@guitar.ucr.edu (john baez) writes...
>>This is another reason why it sort of makes sense to think of negative
>>temperatures as being temperatures that are greater than infinity.
In our thermodynamics course, one lecturer suggested that a 'rational'
system of temperatures might be to define a temperature scale by
T' = -(1/kT) (i.e. -beta?)
In this cases, absolute zero is at - infinity, infinite temperature
is at 0, and 'negative temperatures' are found at positive values. Very nice
in that the value always increases in the 'right' direction, but useless
FMPP/FADTDP. (for most practical/all day to day puroses)
Rather like a thermodynamic equivalent of the 'natural units' of
particle physics or the eminently reasonable step of changing the signs of
electric current and electric charge, really...
- Paul
Johan Wevers
>I thought I'd point out (though it's tangential to the original discussion)
>that gravitational systems do not obey your intuition. A gravitationally
>bound collection of hydrogen, for example, radiates away energy, thus
>collapsing and becoming hotter! (Eventually it becomes a star.)
I don't see this. The temperature increase just arises because of the nuclear
potential energy is converted into motion when a fusion reaction starts.
With big, gaseous planets, like Jupiter and Saturn.
> This
>system has negative heat capacity. Since heat flow then makes hot things
>hotter and cold things colder,
This seems bullshit to me. Take just the potential energy into account.
--
***************************************************************
* J.C.A. Wevers * LaTeX * The only nature of *
* jo...@blade.stack.urc.tue.nl * wizard * reality is physics. *
***************************************************************
john baez
>br...@cco.caltech.edu (David E. Brahm) writes:
>
>>I thought I'd point out (though it's tangential to the original discussion)
>>that gravitational systems do not obey your intuition. A gravitationally
>>bound collection of hydrogen, for example, radiates away energy, thus
>>collapsing and becoming hotter! (Eventually it becomes a star.)
>
>I don't see this. The temperature increase just arises because of the nuclear
>potential energy is converted into motion when a fusion reaction starts.
>With big, gaseous planets, like Jupiter and Saturn.
Yes? This sentence fragment is revealing, since here the
temperature increase is *not* due to fusion.
Brahm was talking about heating *before* the star ignited - it has to
get hotter for ignition to take place in the first place, after all.
Brahm is right, systems in which gravitational binding energy is
significant often have negative specific heats. This is well-known, but
I wish I understood it better, because it seems to point at a deep
connection between gravity and thermodynamics. Does anyone know a
REALLY THOROUGH discussion of this point, which is so important for
understanding the general features of the cosmos (gravitationally bound hot
lumps of matter in cold space, life-forms taking advantage of the
resulting radiation, etc.)?
Groping for a reference (I don't have Zeh's book on me), I yanked out
Reif's nice book Fundamentals of Statistical and Thermal Physics. In
the index there's an entry Specific Heat, positive sign of. Turning to
this, there are only the brief remarks: "It [Positive dE/dT at constant
volume] is a fundamental condition required to guarantee the intrinsic
stability of any phase. This condition is physically very reasonable.
Indeed, the following statement, known as `Le Chatelier's principle,'
must be true quite generally -
If a system is in STABLE equilibrium [Reif's emphasis], then any
spontaneous change of its parameters must bring about processes
which tend to restore the system to equilibrium."
The point, I believe, is that systems in which gravity is significant are
never really in stable equilibrium. They keep crushing down more and
more.
Benjamin Weiner
>br...@cco.caltech.edu (David E. Brahm) writes:
>>I thought I'd point out (though it's tangential to the original discussion)
>>that gravitational systems do not obey your intuition. A gravitationally
>>bound collection of hydrogen, for example, radiates away energy, thus
>>collapsing and becoming hotter! (Eventually it becomes a star.)
>I don't see this. The temperature increase just arises because of the nuclear
>potential energy is converted into motion when a fusion reaction starts.
>With big, gaseous planets, like Jupiter and Saturn.
[adopts cheesy rabbinical accent] Nu, and how does the gas get hot
enough to START the fusion reaction, since it collapsed from a cloud
of molecular hydrogen of temperature ~50 Kelvin?
>> This system has negative heat capacity. Since heat flow then makes
>>hot things hotter and cold things colder,
>This seems bullshit to me. Take just the potential energy into account.
OK, let's do some physics. This argument will be vague and strictly
order of magnitude, but that should be ok since we are arguing about a
sign. Consider a spherical mass of gas (cloud or star) and assume it is
gravitationally bound and (fairly) stable. Then the virial theorem applies:
2 K + W = 0
K is kinetic energy, W grav. potential energy. The cloud has potential
(gravitational binding) energy on the order of
W = - G M^2 / R
where M is its mass and R its size. Here I am neglecting a prefactor
that depends on the density profile ( it's 5/3 for uniform density).
The kinetic energy is keeping the star or cloud from collapsing.
Assume that it is at roughly constant temperature (this is not in
general true but will give us a rough idea.) Then each particle
in the gas has kinetic energy 3kT/2 (assuming monatomicity). Then
K = (3kT / 2) (M / m) = 3 k T M / 2m
where m is the mass of an individual particle (say m = m_proton).
Plugging this into the virial theorem we can solve to get
T = (G M m) / (3 k R), i.e. smaller is hotter.
But also, since the total energy is E = K + W, using the virial theorem
gives E = -K = W/2 . So:
3kM dE
E = - --- T . Since heat cap. C = -- , obviously C < 0 .
2m dT
An astute person will recognize that this has a lot to do with the
all-important minus sign in gravitational potential energy, and that
this is going to be the basis for the statement that the entropy of
a gravitational system is unbounded. To return to David Brahm's
statement, an example of heat flow making hot things hotter and cold
things colder is given by "gravothermal evaporation" in globular
clusters, where the dense core of the cluster becomes progressively
_more_ tightly bound by exchanging energy with the less dense halo
(halo of stars, not dark matter) which becomes less bound.
Doug Merritt
>Indeed, the following statement, known as `Le Chatelier's principle,'
>must be true quite generally -
>
> If a system is in STABLE equilibrium [Reif's emphasis], then any
> spontaneous change of its parameters must bring about processes
> which tend to restore the system to equilibrium."
This is odd because it is a tautology..."stable equilibium" *means*
"positive equilibrium", which in turn means that the system will return
to equilibrium in response to (sufficiently small) perturbations.
Thus this "principle", as stated, means nothing at all.
Perhaps there is another way of stating it that says something about
the resulting processes other than simple tautology.
Doug
--
Doug Merritt do...@netcom.com
Professional Wild-eyed Visionary Member, Crusaders for a Better Tomorrow
DOUGLAS CRAIGEN
>From: jb...@riesz.mit.edu (John C. Baez)
>Subject: Re: below absolute zero?
>Date: Fri, 12 Feb 93 18:15:13 GMT
>In article <1993Feb11....@netcom.com> do...@netcom.com (Doug Merritt) writes:
>>In article <25...@galaxy.ucr.edu> ba...@guitar.ucr.edu (john baez) writes:
>>>Indeed, the following statement, known as `Le Chatelier's principle,'
>>>must be true quite generally -
>>>
>>> If a system is in STABLE equilibrium [Reif's emphasis], then any
>>> spontaneous change of its parameters must bring about processes
>>> which tend to restore the system to equilibrium."
>>
>>This is odd because it is a tautology..."stable equilibium" *means*
>>"positive equilibrium", which in turn means that the system will return
>>to equilibrium in response to (sufficiently small) perturbations.
>>Thus this "principle", as stated, means nothing at all.
>>
>>Perhaps there is another way of stating it that says something about
>>the resulting processes other than simple tautology.
Perhaps you would find the Le Chatelier-Braun Principle more pleasing.
It basically adds to the statement "furthermore, there are also secondary
processes which drive the system towards equilibrium". For example, suppose
you push in on a piston and then release it. The Le Chatelier principle
would tell you that since you have increased the pressure inside compared to
outside, there will be a net force which will push it back. The L-B
principle says there must be more subtle effects as well. This would
encompass a range of effects such as the subsequent expansion of the gas in
the piston causing it to cool slightly, decreasing its pressure.
===================================================================
Doug Craigen, Department of Physics, Acadia University,
Wolfville, N.S., B0P 1X0, (902) 542 - 2201 x150
Da seaweed is always greena
In somebody else's lake
-Sebastian (the crab in "The Little Mermaid")
John C. Baez
>>Indeed, the following statement, known as `Le Chatelier's principle,'
>>must be true quite generally -
>>
>> If a system is in STABLE equilibrium [Reif's emphasis], then any
>> spontaneous change of its parameters must bring about processes
>> which tend to restore the system to equilibrium."
>
>This is odd because it is a tautology..."stable equilibium" *means*
>"positive equilibrium", which in turn means that the system will return
>to equilibrium in response to (sufficiently small) perturbations.
>Thus this "principle", as stated, means nothing at all.
>
>Perhaps there is another way of stating it that says something about
>the resulting processes other than simple tautology.
Well, since I'm a mathematician, everything I ever write is a simple
tautology, so I don't count that as something bad. :-) I agree that
stating the principle (or definition, or whatever) this way makes it
look pretty dull. In the textbook it appears in a section where it is
used to clarify two facts:
1) the heat capacity at constant volume, (dE/dT)_V, must be >= 0,
2) the isothermal compressibility, -(1/V)(dV/dp)_T, must be >= 0
for systems in equilibrium. I think there are lots of other little
facts in chemistry that are also special cases of this tautology.
Physicists often use "tautology" in a somewhat disparaging tone, but I
like 'em 'cause they're true!