Research Article :
The
existence of Crooks fluctuation theorem (even at microscopic level, in a very
short time period) is a direct threat to the second law of thermodynamics. In
this paper, we will underline the flaw that exists in Crooks fluctuation theorem
assumptions, and thus, we will confirm the validity of the second law of
thermodynamics at any temperature, pressure, and at any scale (time, and length-scale)
in nature. We will validate the Loschmidts paradox, and will show that no
physical directional-process can be perfectly-reversible at any non-zero, finite
temperature (T>0 K) and pressure (P>0) in nature. Fluctuation theorem is
a consequence of time reversal symmetry. Gavin E. Crooks and other [1-4] uses
the same theorem in thermodynamics and concluded that there can be a non-zero
possibility during which entropy generation can becomes negatives in short
time-period And thus, they concluded that the second law of thermodynamics does
not hold at nano-scale where work done by the system is the same order as the
available (kBT) thermal energy [2,3].
Gavin E. Crooks assumed that any thermo-dynamical process which happens with
finite speed (could be very small speed, but still it come under finite
category), still this process come under reversible category (he is completely
unaware with this finding) [5]. These researchers think that entropy production
at any time-step can be negative [1-4]. Also, Crooks
[1] proposed that any directional process can be reversible (after using time
reversibility) and can be used to calculate the change in equilibrium Helmholtz
free energy of the system. Crooks argued that if one repeat any thermodynamics
process multiple (ideally infinites) times and then taking time average (using
fluctuation theorems) will provide the change in equilibrium Helmholtz free
energy [1]. Crooks suggested that one can get equilibrium Helmholtz free energy
using time reversal
symmetry, even in ir-reversible process, where work done by the system or
on the system is strictly process (speed, surrounding, polytropic exponent)
dependent. In this
paper, we will show that, fluctuation theorem should not be used in
thermodynamics, because, theorem has inherent flaw in the definition of entropy.
We will show that any directional process (no matter how slow that process has
been performed) has inherent ir-reversibility, and thus, entropy increases in
every time-step during the evolution of the system (which is against the Crooks
fluctuation theorem assumption).We will show that every real system (no dynamic
system can be prefect ideal system due to it interaction with surrounding
during evolution in every time step which induces ir-reversibility in the
process) driven by non-zero finite chemical/electrical/magnetic/gravitational
potential energy gradient which come under ir-reversible category and time
reversal symmetry should not be used to infer any information of the
system at any non-zero temperature (T>0). We will discuss the flaw that
exists in Crooks
fluctuation theorems [1], and in some experimental analysis which have been
done by various experimentalist group, and thus, we will show that the second
law of thermodynamics is still valid in their experiment [2-4]. Before dealing
this problem, we will discuss some very fundamentals rule of thermodynamics
such as ir-reversibility, reversible thermodynamic process and evolution of
process under reversible condition, chemical potential and time reversal
symmetry. We will
start this topic from Carnots statement “A perpetuum mobile is impossible
because of the irreversibility of thermodynamic processes” [6]. The key point
in Carnots statement is “perpetuum mobile”, and “ir-reversibility” of the
thermodynamics process. Question arises that is there any thermodynamics process
in nature which is reversible (in absolute sense)? In this
section, we will validate Carnots statements and will show that there will
never be any thermodynamics process in nature, which will be perfectly
reversible, no matter how slow (even infinitely slow process) one has perform
the process. Before dealing this problem, we will discuss thermodynamics
reversible system. Let see how this process can be performed in real system and
how every real system will be associated with a finite non-zero entropy at every
time-step. Lets take system and surrounding. Both
have the same average-temperature, and same thermodynamic states,
so that reversible thermodynamics process can be performed. We know that the
temperature is macroscopic average quantity. A macroscopic average quantity
always evaluated using statistical rule. So, if there are many thousands of
atoms/molecules, then one can define the average temperature of the system after
adding all possible kinetic
energy of the atoms/molecules (or one can use Maxwell-Boltzmann
distribution curve), and equate it to them . Where, n is the total number of
moles of gas atoms/molecules. This temperature is the average temperature in
investigating volume. This temperature will not be uniform in the entire
volume. There will be some zone in investigating volume, where actual
temperature (depend on the velocity profile of atoms/molecules
(Maxwell-Boltzmann distribution) in that zone) will be bit higher, and also
there will be some zone where actual temperature will be bit lower than the
average temperature. In our analysis, we will use average temperature (mean)
and then we will put temperature error distribution (from mean value) function
which will follow normal-distribution (we have taken this assumption). Now,
lets take our investigating system from thermo-dynamical state A, and move
(reversible) towards the thermodynamics state B. Lets take this process has
completed in infinitely many step, during which system was always in
thermodynamic equilibrium with surrounding (reversible). During this process,
each and every step, we assume that system and surrounding was in thermo-dynamical
equilibrium. Question arises that if both system and surrounding were
always in thermodynamics equilibrium (reversible), then why system goes from
one thermo-dynamical state A, into another thermo-dynamical state B? If system
goes from one thermo-dynamical state A, into another thermo-dynamical state B,
then there must be thermo-dynamical
potential gradient in a very unique direction in which system has evolved.
Presence of any non-zero, finite, (no matter how small this number is, still
this number will be finite in absolute sense) thermodynamic potential gradient
will generate ir-reversibility in the system and, if system have non-zero,
finite (no matter how small this is), entropy at each and every step during the
transition from thermodynamic state A, into thermodynamic state B, then why we
call it reversible process? To answer this question, we will take average
temperature of both system and surrounding at each and every step, as mean
temperature, and then will put error distribution (deviation from
mean-temperature) function of temperature which follows normal distribution.
Lets take average (mean) temperature T0. If we follow normal
distribution in temperature error distribution, then at any time system and
surrounding instant temperatures can be written as T0+δT, and T0-δT,
where δT has been sampled from normal distribution. If one takes average
temperature of both system and surrounding at this time, then, still it is T0
(mean-temperature). Presence of small temperature difference
(2δT) between system and surrounding will allow the differential amount of heat
(δQ) to flow from system into surrounding. We call this δQ is a reversible heat
that has been flown between the system and surrounding at equilibrium (because
heat δQ has been transferred at common temperature (system and surrounding have
common average temperature T0) with reversible process). During this
time, the change in entropy in system, and in surrounding, can be written as, If one analyze this step closely, then one notice that the δQ
amount of heat has been flown out from the system into the surrounding (infinite
bath) without informing the agent (investigator who were watching this step
very carefully during the performance of thermodynamic reversible process). If
one asks with agent that what the entropy change at this-step is in thermo-dynamical
reversible process, then instantly he/she will respond and say that it is
absolutely zero. But, we have just shown that it is not exactly zero. It may be
a very small number, but still non-zero, and finite. If one add this number
infinites times (because whole transition process will take infinites step during
transition from state A, into state B, if it has been performed under
thermo-dynamical reversible process), then it will be very finite and
significant. Now, lets assume that after one step, system temperature become T1,
which is very close to T0. Now, repeat the same process again (now
assume that you have different surrounding which average temperature is also T1).
In this process, the net change in entropy can be written as, (8)where [2] in above
equation, represents the number of time-step, δQ is the amount of heat flow from
system into surrounding in during the thermodynamic reversible process at step
number 2 (we have taken the same amount of heat (δQ) flow in step 1, and in step
2, to make analysis simple), T1 is the system temperature which is very close
to step 1, temperature T0, but not exactly the same, and δT
(positive, assume) is the instant change in temperature between system and
surrounding, which follows the normal distribution. Similar process will be
happened in step number 3, 4, 5,... ∞. Now, the total change in entropy in a
thermodynamic reversible system which transit from state A into state B, can be
written as, Where δQ, is
the amount of differential heat that has been flown from the system into a
surrounding in each time-step under the reversible thermodynamic process, δT,
is the positive temperature difference that has been developed instantly between
system and surrounding and it (δT) follows the normal distribution (we have
assumed this distribution to make our analysis simple. Sometime δT will be
negative, because it has been sampled from normal distribution, but in this
case no heat can flow from system into surrounding, and thus we have ignored
this part and focused only when δT > 0). What we are concluding in this
section is that, if one see the change in entropy (dStotal), in one
time-steps, then, one notice that change in entropy (system +surrounding) is
not absolutely zero during the thermodynamics reversible process. It could be a
very small number, but still it is finite. We generally ignore it and loosely
say that change in entropy during the thermodynamical reversible process is
zero. Since transition
process happens with infinitely many steps (thermodynamical reversible) in
between state A to state B, therefore, the total change in entropy will be very
significant, and thus it will induce ir-reversibility in the process (no matter
how small is this, but still will be non-zero, finite in absolute sense). So no
thermo-dynamical reversible process can be possible without increasing the net
change in entropy of the system+surrounding. In absolute sense, no directional
process (going from state A into state B come under directional process
category) can be perfectly reversible. There will always be a non-zero, finite
ir-reversibility in every process. It does not matter that how slow process has
been performed. Similar process will be happened, if one takes thermodynamics
state B, into state A. In this case, one can take surrounding temperature T0
+δT, and system temperature T0 −δT, and differential heat δQ
will be flown from surrounding into system during single step process. Again, in
this equation T0 is the average temperature (mean-temperature) of
system and surrounding, and δT is the temperature error distribution function
(deviation from mean-temperature), which follows normal distribution (we have
assumed it). Again, the total change in entropy when thermodynamics system goes
from state B, into state A, can be written as (this time define
dStotal=dSsystem+dSsurrounding.
This time heat flows from surrounding into the system with
surrounding temperature T0 + δT, and system temperature T0 −δT) If any
cyclic reversible thermo-dynamical process has been performed between state A,
and state B, then the net change in entropy will be 2Stotal, which is very
small and we simply ignore it and says that change in entropy in a cyclic
process under reversible thermodynamic condition is zero. But, if one arguing
the validity the second law of thermodynamics at any scale, then one must
include this entropy as well (which is non-zero, and finite) in discussion. What we are
concluding in this section is that no directional process can be reversible (in
absolute sense) in nature. It does not matter that how slow that process has
been performed, still there will be a finite ir-reversibility in the process,
and thus, a finite net
positive entropy will be outcome. Since every process in nature is
ir-reversible (in absolute sense), therefore, the second law of thermo dynamics
will be absolute and will always remain valid at any scale (time,
length-scale). To make our discussion simple, we will
use ideal gas equation without losing any generality. In ideal gas, chemical
potential is defined by the Gibbs-Helmholtz equation, Where µ0 and P0
is the chemical potential and pressure at equilibrium, k is the Boltzmann
constant and T is system temperature in Kelvin. As we can see that, chemical
potential µ(T,P), depends on temperature and pressure. Locally, the change in
Gibbs free energy in two different part (space which has different
concentrations, different chemical constituents, different pressure and
temperature) can be written as, depend on the
local differential space denoted by subscript 2, and subscript 1. We have
assumed that the subscript 2 has the higher chemical potential. The net change
in Gibbs free energy will develop the chemical potential
gradient locally between two investigating space, and thus drive the system
(composition of atoms/molecules) either from space 2, towards space 1,or from
space 1, towards space 2. This is a directional process, and thus will come
under ir-reversible category. If system evolves under chemical potential
gradient and goes from investigating space 2, towards space 1, or from space 1,
towards space 2, then, there will be a finite, non-zero, ir-reversible entropy
associated with this process. It does not matter that in which direction
process has been evolved, the entropy will keep adding in previous value, and
thus, keep increasing during the process. If system move in either direction
(due to presence of finite electrochemical potential gradient , which
drive the system in either direction), then the total entropy (system+surrounding)
will increases. One can compare this system with spring-block system, in which
gravitational potential is higher in extreme end, and there is finite gravitational potential
gradient towards the center of the spring-block system (around which simple
harmonic oscillation happen), thus block move toward the center and constantly
interacting with dissipating surrounding media through which ir-reversibility
induces in the system. In this system, entropy
(ir-reversibility) will always increases, and thus spring-block system will
finally come to stop after certain number of oscillation. It does not matter
that block moving towards the center or away from the center. In both cases,
there will be dissipation in energy, and increases in ir-reversibility, and
thus increases in entropy. In other system, it is quite possible that system
has differential non-zero finite chemical potential gradient which is a function
of time, and which change rapidly (because it depends on the concentrations,
pressure and temperature of the constituents) , and thus generates chaotic
motion. Still entropy will increase in each time-step in which system has
driven in certain direction (due to chemical/electrical/magnetic/gravitational potential
energy gradient). The working principle of mitochondria
or any nano-scale devices (where common sense suggest that this process violate
the second law of
thermodynamics, because it is moving in backward direction, and thus one
may assume that it is taking heat from surrounding and converting it into work,
which seems violation of the second law of thermodynamics) can be explained
using chemical potential energy gradient. Chemical potential energy gradient
drives the mitochondria or any nano-scale device in the direction in which the
net chemical potential energy gradient (thermodynamics force). The chemical
potential gradient depends on the temperature, pressure, concentration which is
indirectly dependent on time. If these quantity changes (evolves) with time,
then, chemical potential gradient will also evolves with time (time dependent).
Thus, motion of mitochondria or any nano scale devices
does not violate the second law of thermodynamics at any time and length-scale.
It is general belief in the scientific community that
classical as well as quantum mechanical equations follows time -reversal
symmetry. One should keep in mind that, time flow in one direction only. Also,
every classical and quantum mechanical equation have been derived under very
ideal condition, in which, if one reverse the time arrow, then past
configuration of the system can be obtained (hypothetically. This is also
called Lagrange paradox, because time reversal symmetry will ensure that every
system will be 100% deterministic). So, time reversibility is an ideal concept,
which only valid at zero Kelvin and zero pressure. Because, once system evolves
or move in certain direction under chemical/electrical/magnetic/gravitational,
potential-energy gradient, then there will be a non-zero, finite, ir-reversibility
associated with the system. Because, if any system has been evolved under any
non-zero finite chemical/electrical/magnetic/gravitational, potential
energy-gradient, then this process will no longer called the reversible thermodynamics
process. And, if process is not-reversible, then, there will be a finite
ir-reversibility attached with the system. Also, ir-reversibility always
increases the entropy of system and surrounding. If any system has non-zero
finite ir-reversibility due to presence of chemical/electrical/magnetic/gravitational
potential energy-gradient or due to the effect of surrounding (which is always
present in any real life process), then, one should not invoke the time
reversal symmetry during the analysis of real process. Every real process
always interacts with the surrounding. Interaction with surrounding, and
movement under any kind of chemical/electrical/magnetic/gravitational potential
energy-gradient makes sure that no real-life process can be reversible. Any
thermodynamics process is a real life process, in which thermodynamical system
always interact with the surrounding, and evolves under finite thermodynamics
potential energy-gradient, and thus, every thermodynamics process come under
ir-reversible category. Classical and quantum mechanical
dynamical equations come under ideal category, in which either we ignore
the effect of surrounding, or, we pay no attention towards the evolution of
surrounding. If any system move in either direction, then it will always
interacts and will disrupt the surrounding. If one use time reversal symmetry
in this case, and analyze the system, then disruption on/by the surrounding
will not reverse. Because, disruption produces on/by the surrounding is always
ir-reversible in nature. So, in real-life, no process can be qualified as a
reversible process in absolute sense. Every real-life process is attached with
a non-zero, finite ir-reversibility. Thermodynamics rules (such as second law
of thermodynamics) deals real life process in which system and surrounding
(infinite in size) are coupled to each other, and thus, they are ir-reversible
by nature. Thus, one will not get time-reversible classical or quantum mechanical
equation (none of the equation are time reversible equation in real life,
because surrounding effect (ir-reversibility) always continuously persistent in
one direction only) from thermodynamics. This is exactly the Loschmidts
paradox". In real system, time reversal symmetry can only be invoked at
zero Kelvin, because, at zero Kelvin, disruption due to the surrounding on a
system will be zero. But, at zero Kelvin, everything will be frozen. Nothing
can move in any direction, and thus no thermo-dynamical or real life process
can occurs. So, time reversal symmetry is an ideal concept, which is only valid
at zero Kelvin, and it has no implication in real process. One should not use
time reversal symmetry in real life process, because disruption between system
and surrounding flows in one direction (in time) only. Surrounding always drain
the system, and if system move back in time (using time reversal symmetry),
still then, surrounding will drain the system like it would have drained in
forward time. Dispute between time reversal symmetry
and second law of thermodynamics is very long. Botlzmann himself wrote as soon as one looks at bodies of such small
dimension that they contain only very few molecules, the validity of this
theorem [the second law of thermodynamics and
its description of irreversibility] must cease [3]. However, if one
investigate the microscopic system dynamics carefully, then one notice that
microscopic system evolves under the time varying chemical potential gradient,
which change in every time-step during the evolution. If system evolves under
chemical potential gradient, then there will be a finite ir-reversibility
inherently associated with the process, and thus this process will not be
reversible in absolute sense. So, microscopic process also comes under the ir-reversible
category, and thus this process can not violate the second law of
thermodynamics. Crooks
fluctuation theorem is purely based on key assumption that the microscopic
reversibility is possible at any non-zero finite temperature and pressure
(T>0-K, P>0). It assumes that if any directional-process happens at
infinitely slow speed, then it can be put in a reversible category, which is
inherently wrong [5]. No directional process can be reversible at any non-zero
finite temperature (T>0K), and pressure [5]. It does not matter that how
slow the process has been performed. If process has been performed in certain
direction even at very-very slow speed (infinitely slow), that means there is a
non-zero, finite spatial chemical/electrical/magnetic/gravitational
potential-energy gradient (could be very small, still it come under finite
category) at each and every time-step during the process. If any system has
finite spatial gradient at any temperature and pressure in any process, then,
that process will be a directional process (at each time-step), and this
directional process will generate the finite non-zero entropy in each
time-step. So total, entropy of the system will increase in each and every
time-step. There will be no time-step in which entropy will decreases
[5] it will always increase. Also, time reversibility is an ideal concept,
which is only valid at absolute zero temperature (0-K) and zero pressure. At
absolute zero Kelvin, nothing can be probe for measurement, because, everything
will be completely frozen at zero kelvin. And if one probe for any measurement,
then, locally temperature will rise and thus, system will no longer remain at
absolute zero Kelvin. And, thus, every measurement process under any
temperature and pressure come under directional-process category, which is
itself ir-reversible by nature. Every ir-reversible process generates finite positive entropy
(due to net non-zero finite chemical potential energy-gradient) in each and every
step (no matter in which direction process move, there will always be a
positive finite entropy), and thus, there will never be any step in any process
in which entropy will be decreased. It will always increase [5]. And, thus, we
can conclude that every directional-process or directional-motions either in
classical mechanics or in quantum mechanics at any temperature and pressure
(T>0K,P>0) is ir-reversible by nature (in an absolute sense). (1) Another recently discovered far from equilibrium
expression relates non-equilibrium measurements of the work done on a system to
equilibrium free energy differences. In this paper, we derive a generalized
version of the fluctuation theorem for stochastic, microscopically reversible
dynamics. If any system is far away from the equilibrium, then the work
done by the system, or on the system, cannot be calculated unless or until all
dissipative forces, and polytropic variation of thermo-dynamical variables
are known completely, because, in such cases system always follows irreversible
polytropic process. Also, no stochastic microscopic process is reversible in
absolute sense. If system moves stochastically, then one must investigate each
and every time-step in which system evolves. System can only evolve if there is
a thermodynamic chemical/electrical/magnetic/gravitational, potential-energy
gradient present at each and every time-step. If system moves under any
potential-energy gradient, then process will be ir-reversible by itself, which
will increase the entropy of the system. Also, entropy and time is one way
street. Both move in forward direction only. “Here P(+σ) is the
probability of observing and entropy production rate, σ, measured over a
trajectory of time τ.
This definition of positive and negative entropy production
during forward and in time reversal situation is itself an absurd, because,
there is no time-step in any system where system entropy will decreases. At
each time-step, real system always moves under the presence of locally
chemical/electrical/magnetic/gravitational, potential-energy gradient. Moving
under any potential
energy-gradient will generates ir-reversibility in process, and thus,
increases the entropy. (3) The piston is then
moved inwards at a uniform rate, compressing the gas to some new, smaller
volume. In the corresponding time-reversed process, the gas starts in
equilibrium at the final volume of the forward process, and is then expanded
back to the original volume at the same rate that it was compressed by the
forward process. The microscopic dynamics of the system will differ for each
repetition of this process, as will
the entropy production, the heat transfer, and the work performed on the
system. The probability distribution of the entropy production is measured over
the ensemble of repetitions. If piston move with some finite speed (no matter how small
this speed is), then compression and expansion process will come under
ir-reversible polytropic category [5]. If process is ir-reversible and
polytropic then work done during compression and expansion cannot be calculated
unless or until all dissipative forces and exponent ƞ, is known completely.
Also, due to presence of dissipative forces, this work will be the path and
speed dependent. So, calculating entropy or any other thermodynamic variable
under this condition after multiple repetition of the experiment has no solid
thermodynamic foundation backing, and thus, cannot be trusted without any
doubt. (4) Another expression
that is valid in the far-from equilibrium regime is the recently discovered
relationship between the difference in free energies of two equilibrium ensembles,
δF, and the amount of work, W, expended in switching between ensembles in a
finite amount of time Here KB is
the Boltzmann constant, T is the temperature of the heat bath that is coupled
to the system, and indicates an average over many repetitions of the switching
process". Strictly speaking, no direct relation can be obtain between Helmholtz free energy
and work done in a ir-reversible process, until or unless all dissipative
forces which is speed dependent, and polytropic exponents of gas is known
completely. Because, work done in an ir-reversible process will be path
dependent. Any process which has been performed in a finite time will come
under ir-reversible category. (5) From state A to
state B during some finite time interval. The stochastic dynamics that are
typically used to model reversible physical systems coupled to a heat bath,
such as the Langevin equation and Metropolis Monte Carlo, are microscopically
reversible in the sense of Equation [5]. Generally, if the dynamics of a system
are detail balanced locally in time (i.e., each time step is detail balanced),
then the system is microscopically reversible even if the system is driven from
equilibrium by an external perturbation. Langevin
equation, Has dissipative term ( ) which is velocity ( ) dependent. Presence of dissipative
term will make whole equation ir-reversible which will always increase entropy.
So, if one run this equation back in time, then dissipative term behaves
exactly the same way as it would have behaved when time flow in forward
direction, and thus, increases the ir-reversibility. Thus entropy will keep
increasing and flow in one direction, like time flow in real life. Real system evolves under the presence of
chemical/electrical/magnetic/gravitational, potential-energy gradient at each
time-step, and thus Metropolis Monte Carlo must include ir-reversibility term
in dynamic equation, otherwise this equation will never represent any real
thermo-dynamical system, and thus, no final conclusion on entropy and second
law of thermodynamics should have been made using Metropolis Monte Carlo
outcome until or unless it represent the real system. (6) Clearly there is a
change in entropy due to the exchange of energy with the bath. If Q is the
amount of energy that flows out of the bath and into the system, then the
entropy of the bath must change by –βQ. Strictly speaking, this statement is only correct when heat
Q, has been exchanged under reversible condition (reversible condition is an
ideal concept. It requires infinitely slow speed to perform any work. Under
reversible condition, infinite amount of time will take to complete any
process. If any process happens in finite amount of time, then it cannot be
called reversible process). If heat Q, has been exchanged under ir-reversible
condition (as in Crooks case), then total entropy will be higher than –βQ, due
to the ir-reversibility contribution. And then change in entropy should be
equal to –βQ-–βWlost [7]. (7) For example, with
the confined gas we compare the entropy production when the gas is compressed
to the entropy production when the gas is expanded. To allow this comparison of
forward and reverse processes, we will require that the entropy production is
odd under a time reversal, i.e., wF = -wR , for the
process under consideration. Expansion and
compression of a gas under ir-reversible condition is path, polytropic
constant, and speed dependent. This dissipative process should not be made time
reversible, because dissipative forces will always dissipate energy and
increase entropy in every time-step. It does not matter in which direction
process move entropy will keep increasing in one direction only. One can
compare this process with a toll-tax collection done by some government/private
agent on a road. If one use road and goes in one direction, then toll-tax has
to be paid, and if once come back again on the same road then again toll-tax
has to be paid. The total toll tax in one, two-way trip will be the double of
one-way trip. Here toll-tax plays the same role as entropy play in directional
process. If one reverses time, still entropy will increases from previous
value. It will never decrease, because entropy and time (in real situation) is
only one way street (both move in forward direction only). (8) Both baths are
considered to be large, equilibrium, thermodynamic systems. Therefore, the
change in entropy of the heat bath is –βQ and the change in entropy of the
volume bath is -PδV where delta δV is the change in volume of the system. This statement is only valid if heat, Q, and work, PδV, has
been performed under thermo-dynamical reversible condition (which will take
infinite amount of time). If process has been performed with non-zero speed and
completed in finite time, then above statement is wrong. Because entropy
contribution due to ir-reversible process must be included (every process is
ir-reversible process in real life. Only degree of ir-reversibility varies
between processes to process, but no process can be 100% reversible). (9) We start from the
appropriate non-equilibrium
steady state, at a time symmetric point of λ(t), and propagate forward in
time a whole number of cycles. The corresponding time-reversed process is then
identical to the forward process, with both starting and ending in the same
steady-state ensemble. The entropy production for this system is odd under a
time reversal and the fluctuation theorem is valid. No two ir-reversible process can be identical either in
forward or in backward directions. Fluctuation theorem is
not valid in thermodynamics. There is very inherent flaw in entropy definition
itself in fluctuation theorem. Entropy can never be negative at any time-step.
Thermo-dynamical system evolves under the chemical/electrical/magnetic/gravitational
potential-energy gradient. If any system evolves under the presence of any
gradient, then there will be inherent ir-reversibility associated with that
process. (10) The dynamics are
required to be stochastic, Markovian, and microscopically reversible, equation
(5), and the entropy production, defined by equation (6), must be odd under a
time reversal. Microscopic dynamics is driven by
chemical/electrical/magnetic/gravitational potential-energy gradient.
Microscopic system evolves under the presence of time varying finite chemical
potential-energy gradient, and ir-reversibility keep getting accumulated in
each and every time-step, and thus, no microscopic dynamics
is reversible in absolute sense. Also, entropy always increases in ir-reversible
process, and it is only one way street. Reversal of entropy is not possible by
any means in any directional process which has been evolved under non-zero,
finite chemical/electrical/magnetic/ gravitational, potential-energy gradient.
Every real life system only evolves on the presence of any type of
potential-energy gradient (time dependent/independent driving force) and thus
inherently ir-reversible. (1) GM
Wang, et al. wrote, we experimentally
demonstrate the fluctuation theorem, which predicts appreciable and measurable
violations of the second law of thermodynamics for small systems over short
time scales, by following the trajectory of a colloidal particle captured in an
optical trap that is translated relative to surrounding water molecules. From
each particle trajectory, we calculate the entropy production/consumption over
the duration of the trajectory and determine the fraction ofsecond law defying trajectories. Our
results show entropy consumption can occur over colloidal length and time
scales [2]. The key point in above
statement is that, G.M. Wang et al has observed that colloidal particle deed
second law of thermodynamics, and it move in opposite direction against the
prediction of second law of thermodynamics. Question arises that on what
fundamental ground G.M. Wang et al. have decided the direction of the second
law of thermodynamics? The rule of thermo-dynamics says that it must be decided
only after the careful evaluation of chemical/electrical/ magnetic/
gravitational potential-energy gradient. In G.M.Wang et al. experiment,
colloidal particle will move in the direction in which chemical potential
energy gradient will dictates. The time varying (due to external perturbation)
chemical potential energy depends on temperature, pressure and concentration of
colloidal particles in his experiment. Question is, did G.M. Wang et al. have
included these key gradients in his analysis? Answer is absolutely no. Still,
his team has the audacity to conclude the violation of the second law of
thermodynamics in his experiment. It happens due to the pure ignorance effect. (2) If the work
performed during the duty cycle of any machine is comparable to thermal energy
per degree of freedom, then one can expect that the machine will operate in
reverse over short time scales. That is, heat energy from the surroundings will
be converted into useful work allowing the engine to run backwards. For larger
engines, we would describe this as a violation of the
second law of thermodynamics, as entropy is consumed rather than generated [2].
Machine will operate only in that direction in which there
will be finite chemical potential gradient. Chemical potential energy depends
on the temperature, pressure, and concentrations of atoms/molecules locally
where machine is situated. Chemical potential energy gradient will decide the
direction in which machine will move. This motion is well accordance with the
second law of thermodynamics (in-fact this directional movement will be
dictated by the second
law of thermodynamics itself). Moving under finite chemical potential
energy gradient will generate ir-reversibility and thus increase finite
entropy. No process can consume entropy in any time-step. (3) The only thermodynamic
statement available was the second law itself, stating that, for large systems
and over long times, the entropy production rate is necessarily positive. Even
the foundations of statistical mechanics were unsettled as thermo-dynamists
questioned how the second law of thermodynamics could be reconciled with
reversible microscopic equations of motion. Loschmidts paradox states that in a
time reversible system, for every phase-space trajectory there exists a
time-reversed anti- trajectory [1]. As the entropy production of a trajectory
and its conjugate anti-trajectory are of identical magnitude but opposite sign,
then, so the argument goes, one cannot prove that entropy production is
positive [2]. The second law of thermodynamics only says that the evolution
of any thermo-dynamical system will generate ir-reversibility and thus increase
entropy. The second law of thermodynamics does not distinguish between
small/large space dimension and time. The second law of thermodynamics is
applicable to each and every time-step and any length-scale. Any real
microscopic process is not reversible in any time-step and length-scale.
Microscopic system drives under the chemical potential energy gradient, which
is ir-reversible. It is only reversible in an imaginary computer simulation, in
which outcome of the simulation has nothing to do with the real practical
problems which happens in nature. No real process, either classical, or quantum
mechanical is time-reversible at any time-step and length-scale. Classical and quantum mechanical
equations are the ideal equation, and only applicable at the zero Kelvin.
But, then, everything will be frozen at zero Kelvin and thus, no dynamics is
possible. There is no real system in which the entropy can be absorbed during
it evolution, because, evolution of any system is driven by
chemical/electrical/magnetic/gravitational potential energy gradient, which is
inherently ir-reversible. (4) That
entropy-consuming trajectories can be discerned for micron-sized particles over
time scales on the order of seconds is particularly important to applications
of nano-machines and to the understanding of protein motors. The fluctuation
theorem points out that as these thermodynamic engines are made smaller and as
the time of operation is made shorter, these engines are not simple scaled-down
versions of their larger counterparts. As they become smaller, the probability
that they will run thermodynamically in reverse inescapably becomes greater.
Consequently, these results imply that the fluctuation theorem has important
ramifications for nanotechnology and indeed for how life itself functions [2]. There is no trajectory in which entropy can get consumed.
Because, every trajectory has been evolved under non-zero, finite chemical
potential gradient, which inherently increases the ir-reversibility in the
process and thus entropy, increases in each and every time-step. Thermodynamics
process run under chemical potential energy gradient, which depends on
temperature, pressure and concentrations of atoms/molecules. Chemical potential
energy gradient will decides the direction in which thermo-dynamical process
will evolves. This evolution will be dictated by the finite potential energy
gradients, which is nothing but the second law of thermodynamics itself. Carberry DM, et al. wrote [6], (1) The puzzle of how
time-reversible microscopic equations of mechanics lead to the
time-irreversible macroscopic equations of thermodynamics have been a paradox
since the days of Boltzmann. Boltzmann simply sidestepped this enigma by
stating as soon as one looks at bodies of such small dimension that they
contain only very few molecules, the validity of this theorem [the second law
of thermodynamics and its description of irreversibility] must cease."
Today we can state that the transient fluctuation theorem (TFT) of Evans and
Searles is a generalized, second-law-like theorem that bridges the microscopic
and macroscopic domains and links the time-reversible and irreversible
descriptions. We apply this theorem to a colloidal particle in an optical trap.
For the first time, we demonstrate the TFT in an experiment and show
quantitative agreement with Langevin dynamics [3].
There is no time-reversible process exist in nature at any
temperature T>0K. Fluctuation theorem has a very inherent flaw in defining
the entropy. Under no circumstances entropy can be negative in any directional
process at any-time steps. Therefore, fluctuation theorem will never be able to
represents the second law of thermodynamics. If colloidal particles follows the
Langevin dynamics, Then how is the colloidal motion challenges the second law of
thermodynamics? Because, Langevin
equation is inherently ir-reversible in nature due to presence of
dissipative term (
) which depends on the speed of the colloidal particle. If one reverses
the dynamics of the Langevin equation back in time, then, this dissipative term
will keep dissipating energy and will act like as process moving in the forward
in time. Presence of this dissipative term will drive the process in
ir-reversible direction. Also, no real process is reversible in nature in
absolute sense. (2) The second law of
thermodynamics states that for systems in the thermodynamic limit, the entropy
production must be greater than or equal to zero. Although the underlying
equations of motion are time reversible, the second law predicts an
irreversible entropy production. This law applies to systems that are of
infinite size and persist over long times. Despite this strict limitation, the
second law is often treated as being universal in application because the size
of most systems can be considered infinite when compared to atomic length and
time scales. However, several systems of current scientific interest, such as
nano-machines and protein motors, operate at length and time scales where the
system cannot be considered infinite. At the nano and microscales the thermal energy
available per degree of freedom can be comparable to the work performed by the
system. Classical thermodynamics does not apply to these small systems [3]. No directional motion is reversible in nature in absolute
sense. Every directional process is driven by the chemical potential energy
gradient, which is inherently ir-reversible by nature. The second law of
thermodynamics is applicable to any system at any scale. This law only says
that if any system evolves under chemical/electrical/magnetic/gravitational
potential energy gradient, then the ir-reversibility of the process increases
which will increases the entropy of the system. Even at the nano-scale, system
evolves only under the presence of the chemical potential energy gradient, and
thus, follows the second law of thermodynamics. (3) In this way the FT
can be viewed as a generalization of the second law since the FT applies to
finite systems observed over finite time and trivially recovers the second law
in the thermodynamic limit [3]. There is a fatal inherent flaw in the fluctuation theorem.
Entropy can never be negative in any directional process. Entropy and time is
only one way street. Both continuously flow in forward direction. Fluctuation
theorems will never be able to represent the second law of thermodynamics
any-day, anytime. (4) If the probability of observing a trajectory in δV is
equal to the probability of observing one in δV*, then the system is
thermodynamically reversible and Ωt=0 [3]. Thermodynamic reversibility is
always defined between system and surrounding when process perform infinitely
slow. Every thermodynamic variables, such as, temperature, pressure, Gibbs free
energy, chemical potential, are in equilibrium between system and surrounding
(there is no any type of spatial-gradient). Thermodynamic reversibility concept
is not allowed if any system evolves under the non-zero, finite gradient
(any-type). Tracing of any directional trajectory back in time after using time-reversal symmetry
does not come under the thermodynamics reversible category. (5) The TFT
quantitatively describes how irreversible macroscopic behavior evolves from
time-reversible microscopic dynamics as either the system size or the
observation time increases. This experiment confirms, for the first time, the
predictions of this theorem. Furthermore, these experiments demonstrate that
the effects predicted occur over colloidal length and time scales and,
consequently, show that the TFT is relevant to nano-technological
applications [3].
There is no reversible process in nature at any finite
temperature (T>0K) in an absolute sense. Experiment confirms nothing about
TFT. But, it does confirmed the validity of the second law of thermodynamics if
one analyze the system and surrounding carefully, using the chemical potential
energy gradient concept. FT/TFT will never be able to evaluate the exact change
in a entropy of any system due to inherent fatal flaw that exists in the
definition of entropy in FT/TFT. Collin D, et al. wrote, (1) A consequence of the
CFT is Jarzynski equality, which relates the equilibrium free-energy difference
dG between two equilibrium states to an exponential average
(denoted by angle brackets) of the work done on the system, W, taken over an
infinite number of repeated none-equilibrium experiments [4], The work done in an infinite numbers of repeated non-equilibrium
experiment come under the ir-reversible category [5]. However, standard
thermodynamics fundamentals suggest that, work must need to be performed under
reversible process, (reversible process will take infinite amount of time for
one cycle) only then work W and G can be equated. If process happens with
infinite speed, or infinite time, then, it come under the irreversible
category, and ir-reversible process rule needs to be invoked before reaching on
any conclusion [5]. (2) These results
illustrate that when used in con-junction with an appropriate fluctuation
theorem, non-equilibrium
single-molecule force measurements can provide equilibrium information such
as folding free energies, even if the process studied occurs under
far-from-equilibrium conditions [4]. The ir-reversible forces will never be able to provide the
any information about the equilibrium thermodynamics variables. Also, unfolding
and folding process of RNA molecules should have been explained using the chemical
potential energy gradient, which depends on the temperature, pressure and the
local chemical compositions of atoms/molecules. Fluctuation theorem has very inherent fatal flaw in the
definition of entropy itself. Entropy will never become negative in any
directional process in any time-step and length-scale. In nature, entropy and
time is one way street, both flow in forward direction only. In this paper, we have addressed the flaw that exist in
Crooks fluctuation theorems. We have shown that the every directional process
in nature evolves under the chemical/electrical/magnetic/gravitational
potential energy gradient, which is inherently ir-reversible by nature,
therefore, no reversible process exists in nature in absolute sense at any
finite temperature. We have shown that the evolution of microscopic process is
ir-reversible due to the presence of directional chemical potential energy
gradient at each time-step, which drive the entire system in one direction. We
have shown that if any process has been performed with infinite speed and in
infinite time, then it will not come under the reversible process category. We
have shown that fluctuation theorem has very inherent fatal flaw in the
definition of entropy, and it will never be able to capture the second law of
thermodynamics. We have shown that entropy and time are only one way Street, in
which both will continuously increase and move in one direction only. We have
shown that experiment in which the second law of thermodynamics violation has
been claimed have inherent aw in data analysis. We have shown that all
experimented data can be explained using chemical potential energy gradient. We have shown that the system will be evolved and governed by
the chemical potential gradient and thus irreversibility will induce in the
process itself which will increase entropy. We have shown that evolution of
system under chemical potential energy gradient in each time step is governed
by the second law of thermodynamics. We have shown that no directional process
which have been evolved or will evolve can come under reversible category. We
have discussed the reversible
thermodynamics and its evolution which ultimately increases the non-zero
entropy. We have shown that the every process which operates at any non-zero
temperature (T>0K) is inherently ir-reversible by nature, and thus
fundamental thermodynamic quantity such as entropy will always increases in
each and every time-step. We have shown that in nature entropy increases in
every time-step and thus, validate the supremacy of the second law of
thermodynamics. We hope that this paper will clear the doubt which surrounds
the second law of thermodynamics, and thus, stimulate the scientific community
to start looking the fundamentals thermodynamics equations at microscopic
levels very carefully. Now, we can quote Arthur Eddington again The second law of
thermodynamics holds, I think, the supreme position among the laws of Nature.
If someone points out to you that your pet theory of the universe is in
dis-agreement with Maxwells
equations then so much the worse for Maxwells equations. If it is found to
be contradicted by observation, well, these experimentalists do bungle things
sometimes. But if your theory is found to be against the second law of
thermodynamics I can give you no hope, there is nothing for it but to collapse
in deepest humiliation [6]. 1.
Gavin
CE. Entropy production fluctuation theorem and the non-equilibrium work
relation for free energy differences (1999) Phy Rev E 60: 2721-2726. 2.
Wang
GM, Sevick EM, Mittag E, Searles DJ and Evans DJ. Experimental demonstration of
violations of the second law of thermodynamics for small systems and short time
scales (2002) Phys Rev Lett 89: 050601. 3.
Carberry
DM, Reid JC, Wang GM, Sevick EM, Searles DJ, et al. Fluctuations and irreversibility:
An experimental demonstration of a second-law-like theorem using a colloidal
particle held in an optical trap (2004) Evans Phy Rev Lett 92: 14060. 4.
Collin
D, Ritort F, Jarzynski C, Smith SB, Tinoco I, et al. Verification of the crooks
fluctuation theorem and recovery of RNA folding free energies (2005) Nat lett
437: 231-234. 5.
Petrescu
S, Borcila B, Costea M, Banches E, Popescu G, et al. Concepts and fundamental
equations in Thermodynamics with Finite Speed (2016) IOP Conference Series:
Materials Science and Engineering, UK 147: 012144. 6.
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A and Sheehan D. Quantum limits to the second law of thermodynamics (2004)
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https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node48.html Gokaran
Shukla, School of
Physics, Trinity College, Dublin 2, Ireland, E- mail: shuklag@tcd.ie Shukla G. Flaw in crooks
fluctuation theorem (2018) Edelweiss Chem Sci J 1: 13-20 Fluctuation theorem, Thermodynamics, Time reversal symmetry,
Thermal energyFlaw in Crooks Fluctuation Theorem
Abstract
Full-Text
Introduction
Ir-Reversibility in Nature
Chemical Potential
Time Reversal Symmetry
Key Points in Crooks Fluctuation Theorem [1]
Shortcoming in crooks fluctuation theorems
[1]
There are many short coming in Crooks fluctuation theorem. We will go one
by one. Gavin E. Crooks wroteShortcoming in experimental analysis [2-4]
Conclusion
References
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Keywords
