chalidze
Irreversibility -- a defense against Laplace's Demon
Thermodynamics was created to explain the behavior of thermal machines and
the like, not to clarify the philosophy of free will. Yet, it comes in
handy that there is a function in thermodynamics -- entropy -- that
characterizes the irreversibility of processes in a physical system. To
explain this property, Boltzman developed statistical mechanics which, in
his opinion, connected entropy with the movement of then hypothetical
particles: atoms and molecules. Yet in classical
physics the movement of atoms is mechanical, therefore reversible. How
does one explain that the development of a gaseous system is often
irreversible, with disorder increasing and entropy growing?
Loschmidt declared it to be impossible, using the following trick for
illustration. In an isolated system with growing entropy from the moment
T1, let's imagine that at the moment T2 the velocities of all particles
are reversed. Then from that moment entropy will be decreasing and at some
moment, T3, entropy will reach the initial lower level. That is a
violation of the Second Law Of Thermodynamics.
The plausible explanation would be that the state of gas with reversed
velocities at moment T2 cannot be realized in Nature, as well as all
states between moments T2 and T3. That are too many forbidden states with
no evidence that systems have any way to know which states are forbidden.
For many years I myself felt the same dissatisfaction with Boltzman's
H-theorem as with Godel's theorem of incompleteness, but science is not
supposed to be built on the basis of feelings.
The connection between my search for how to escape the totalitarian
control of Laplace's Demon, and the irreversibility in a statistical
system, is clear: if movements and ideally elastic collisions of atoms are
defind by mechanical laws, one would expect that they must be reversible,
and the Loschmidt trick is the way to check it.
But here is a problem with mechanical equations which, as noted before,
generally provide power to the Laplacian Demon: generally, equations cannot describe
the result of the simultaneous collision of three or more atoms. Free
movement and double collisions -- yes, we have solutions for that, but not
multiple collisions.
In empty space with atoms flying in all directions, movement is defined.
In not dense gas with only double elastic collisions, it is also defined
and completely reversible. One cannot build a useful thermal engine based
on such a system, because the engine irreversibly takes energy from the
system for our use.
However, if we have multiple non-symmetrical collisions, then we have
irreversibility (symmetry can give us ways to calculate it in some cases).
Here is how it works. At moment T1, four atoms in a non-symmetrical
configuration are heading to collide at once at a certain point. T2 is the
moment of collision. At moment T3 they fly apart, and we generally cannot
calculate the direction and velocity of each atom.
If after the moment T3 we apply the Loschmidt trick, the atoms will go
back to the point of collision, but generaly speaking they will not get
back to the state in which they were at moment T1. Those four atoms simply
"forgot" where they were before the collision. This is a loss of
information. This is irreversibility, real and without violating any
mechanical laws, i.e., conservation of energy, momentum and moment of
momentum. There are simply not enough mechanical laws to define movement
after multiple collisions. Such situations are called singularities.
Leaving aside the obvious case of non-elastic collisions, what can be the
reason for growing entropy in a system with chaotic collisions? Is entropy
connected with singularities? It is easy to suppose so, especially because
the processes we study are not unraveling in empty space but in vessels
with walls, and those walls by themselves must be a great source of
singularities. Take the piston of a machine, for example. It contains many
of its own particles, and they are all interdependent as particles are in
solid bodies. In kinetic theory particles are presumed to be independent;
therefore, the piston of a steam engine in relation to atoms of gas is a
separate particle itself, as well as the wall of the vessel. A multitude of atoms collide with that piston and the walls, creating
singularities moment after moment.
Any gas with thermal movement can be characterized by the quantity of
singularities per second -- the function of irrevercibility A (for atoms'
amnesia about their history before the last multiple collision). For gas
in empty space with atoms flying without collision, or with only double
collisions, A=0. For gas in a vessel, A>0 due to singularities produced by
collisions with the walls of the vessel, and multiple collisions of atoms.
It grows with temperature, it grows with the number of atoms per unit of
volume. Function A is a more natural characteristic of irreversibility
because the production of singularities is purely a mechanical process:
either multiple collisions occurred, or not. Entropy, on the other hand,
is a still somewhat mysterious probabilistic characteristic. (When I was
younger I wrote a book with the arrogant title "Entropy Demystified".
Well, it is still a mystery.) And then there are some shadows of paradoxes connected with subjectivity.
Should I celebrate the freedom of the world from Laplace's Demon? Indeed,
the Demon does not know where this or that atom will go after multiple
collision. Yet it feels more like a tragedy for my personal world's
perception: if an atom goes to point P without being directed there and
only there by mechanical laws, what is left of my belief in world order?
What will happen with the intellectual citadel in my brain, the belief
that if something is happening, there must be reason for that? Therefore,
my inner world-policeman is whispering: maybe there are many more
mechanical laws than we know, and they provide for more equations; maybe
all atoms after singularities know precisely where to go? This is
unlikely.
The last chance to preserve the illusion of world order is to say that
there is an impossibility that more than two atoms will collide at the
same moment, but one must be really desperate to state this. And I would
be the last one to do so, because I recognize the reality of time only in
connection with life, not in the inanimate world.
Some problems might exist due to the fact that I only found escape from
the Demon for gases with chaotic collisions. Is that also applicable for
liquid crystal, like the medium inside the living cells where the main
life processes take place? I don't see why not, but one might choose to
analyze it more carefully.
Celebrate or not, at least thanks to singularities we know so far that the
physical order of the world is not an obstacle for freedom of will. We
have room to act by our choice without violating the laws of physics.
These notes on irreversibility are even rougher than certain other of my
notes, but I decided to put them here as I am going to refer to them
before polishing, if ever.
Valery Chalidze. Dec 6, 2010
Thermodynamics was created to explain the behavior of thermal machines and
the like, not to clarify the philosophy of free will. Yet, it comes in
handy that there is a function in thermodynamics -- entropy -- that
characterizes the irreversibility of processes in a physical system. To
explain this property, Boltzman developed statistical mechanics which, in
his opinion, connected entropy with the movement of then hypothetical
particles: atoms and molecules. Yet in classical
physics the movement of atoms is mechanical, therefore reversible. How
does one explain that the development of a gaseous system is often
irreversible, with disorder increasing and entropy growing?
Loschmidt declared it to be impossible, using the following trick for
illustration. In an isolated system with growing entropy from the moment
T1, let's imagine that at the moment T2 the velocities of all particles
are reversed. Then from that moment entropy will be decreasing and at some
moment, T3, entropy will reach the initial lower level. That is a
violation of the Second Law Of Thermodynamics.
The plausible explanation would be that the state of gas with reversed
velocities at moment T2 cannot be realized in Nature, as well as all
states between moments T2 and T3. That are too many forbidden states with
no evidence that systems have any way to know which states are forbidden.
For many years I myself felt the same dissatisfaction with Boltzman's
H-theorem as with Godel's theorem of incompleteness, but science is not
supposed to be built on the basis of feelings.
The connection between my search for how to escape the totalitarian
control of Laplace's Demon, and the irreversibility in a statistical
system, is clear: if movements and ideally elastic collisions of atoms are
defind by mechanical laws, one would expect that they must be reversible,
and the Loschmidt trick is the way to check it.
But here is a problem with mechanical equations which, as noted before,
generally provide power to the Laplacian Demon: generally, equations cannot describe
the result of the simultaneous collision of three or more atoms. Free
movement and double collisions -- yes, we have solutions for that, but not
multiple collisions.
In empty space with atoms flying in all directions, movement is defined.
In not dense gas with only double elastic collisions, it is also defined
and completely reversible. One cannot build a useful thermal engine based
on such a system, because the engine irreversibly takes energy from the
system for our use.
However, if we have multiple non-symmetrical collisions, then we have
irreversibility (symmetry can give us ways to calculate it in some cases).
Here is how it works. At moment T1, four atoms in a non-symmetrical
configuration are heading to collide at once at a certain point. T2 is the
moment of collision. At moment T3 they fly apart, and we generally cannot
calculate the direction and velocity of each atom.
If after the moment T3 we apply the Loschmidt trick, the atoms will go
back to the point of collision, but generaly speaking they will not get
back to the state in which they were at moment T1. Those four atoms simply
"forgot" where they were before the collision. This is a loss of
information. This is irreversibility, real and without violating any
mechanical laws, i.e., conservation of energy, momentum and moment of
momentum. There are simply not enough mechanical laws to define movement
after multiple collisions. Such situations are called singularities.
Leaving aside the obvious case of non-elastic collisions, what can be the
reason for growing entropy in a system with chaotic collisions? Is entropy
connected with singularities? It is easy to suppose so, especially because
the processes we study are not unraveling in empty space but in vessels
with walls, and those walls by themselves must be a great source of
singularities. Take the piston of a machine, for example. It contains many
of its own particles, and they are all interdependent as particles are in
solid bodies. In kinetic theory particles are presumed to be independent;
therefore, the piston of a steam engine in relation to atoms of gas is a
separate particle itself, as well as the wall of the vessel. A multitude of atoms collide with that piston and the walls, creating
singularities moment after moment.
Any gas with thermal movement can be characterized by the quantity of
singularities per second -- the function of irrevercibility A (for atoms'
amnesia about their history before the last multiple collision). For gas
in empty space with atoms flying without collision, or with only double
collisions, A=0. For gas in a vessel, A>0 due to singularities produced by
collisions with the walls of the vessel, and multiple collisions of atoms.
It grows with temperature, it grows with the number of atoms per unit of
volume. Function A is a more natural characteristic of irreversibility
because the production of singularities is purely a mechanical process:
either multiple collisions occurred, or not. Entropy, on the other hand,
is a still somewhat mysterious probabilistic characteristic. (When I was
younger I wrote a book with the arrogant title "Entropy Demystified".
Well, it is still a mystery.) And then there are some shadows of paradoxes connected with subjectivity.
Should I celebrate the freedom of the world from Laplace's Demon? Indeed,
the Demon does not know where this or that atom will go after multiple
collision. Yet it feels more like a tragedy for my personal world's
perception: if an atom goes to point P without being directed there and
only there by mechanical laws, what is left of my belief in world order?
What will happen with the intellectual citadel in my brain, the belief
that if something is happening, there must be reason for that? Therefore,
my inner world-policeman is whispering: maybe there are many more
mechanical laws than we know, and they provide for more equations; maybe
all atoms after singularities know precisely where to go? This is
unlikely.
The last chance to preserve the illusion of world order is to say that
there is an impossibility that more than two atoms will collide at the
same moment, but one must be really desperate to state this. And I would
be the last one to do so, because I recognize the reality of time only in
connection with life, not in the inanimate world.
Some problems might exist due to the fact that I only found escape from
the Demon for gases with chaotic collisions. Is that also applicable for
liquid crystal, like the medium inside the living cells where the main
life processes take place? I don't see why not, but one might choose to
analyze it more carefully.
Celebrate or not, at least thanks to singularities we know so far that the
physical order of the world is not an obstacle for freedom of will. We
have room to act by our choice without violating the laws of physics.
These notes on irreversibility are even rougher than certain other of my
notes, but I decided to put them here as I am going to refer to them
before polishing, if ever.
Valery Chalidze. Dec 6, 2010