EXTROPY, ELrlSTIC STRAIIL’, AND T H E SECOND LAW O F THERMODYKAMICS; T H E PRIKCIPLES O F LEAST WORK AND OF MAXIMUM PROBABILITY BY TV. S . KIMBALL
The purpose of this paper is to point out a new mechanical aspect of entropy with special reference to simple fully excited gas. By relying on the geometrical expression for weight, W = SN(rl . . . . r N ) and taking strains to include unit extensions in velocity and momentum space as well as ordinary space, a relation is established between entropy and the total strain: S = kY; Y = log W,where S is the entropy, k is Boltzmann’s constant, Y the total strain, and W is the a-priori probability. The Lagrange multiplier method is applied to statically indeterminate frames as well as gas theory, and both cases show that the Principle of Least Work, or Internal Energy is equivalent to the Principle of Maximum Entropy, Strain and a-priori probability. Furthermore the equations which by the Lagrange method determine the equilibrium state are shown to represent balance between true stress and strain both in gas theory and in the statics of indeterminate frames, revealing the operation in these two domains of the same identical principles (not an analogy). And the modulus of elasticity for these stresses and strains acting in momentum space is the ordinary bulk modulus p (the pressure) that also applies to volume expansions in ordinary space. Then the second law of thermodynamics and the automatic increases of entropy that it represents are explained as due to increased strain (in momentum space as well as ordinary space) under action of corresponding stresses, rather than as in statistics by the unsatisfactory ergodic hypothesis. Altho these ideas are radicai in that the statistical aspect is purposely put aside in favor of the geometrical and mechanical aspect, yet they have classical support from Boltzmann’s H-theorem which shows by the method of collisions (and the forces involved) that the rate of change of entropy is positive. On the other hand Boltzmann’s H-theorem implying the operation of forces to bring about the increase of entropy towards equilibrium, is in disagreement with any statistical theory that requires equilibrium to be reached without operation of forces, solely from probability considerations and the ergodic hypotheses. I t seems very possible that equilibrium between stresses and strains may accompany and account for various other phenomenon of gaseous equilibrium such as fluctuations and Brownian motion that usually are treated by the statistical method, and are not within the scope of the present article. Since the geometrical weight method has already been applied to the new quantum statistics,’ it seems likely that the present mechanical stress-strain theory that comes from the geometrical weight method, can also be extended to include the new statistics. ~
1
Chandrasekhar: Phil. Mag., 9, 621 (1930).
W. S. KIMRALL
612
1. The Geometrical Expression for Weight
and the Statistical
Expression for Weight The familiar expression for weight used in statistical mechanics is :
JV
=
N! (wl)”’(wz)”’. nl!n2! . . . .nk!
.
. .(Uk)”k
(1)
where the w’s are small volumes in “p-space” (of 6 to 1 2 dimensions according as we refer to molecules with three or six degrees of freedom). And their product (when each is raised to the indicated power) gives the elementary volume in so called y-space, which is known as the geometrical part of the weight given by ( I ) . The numerical coefficient involving the factorials of the n’s is the statistical part and indicates the number of distinct arrangements in this y-space. This expression for weight ( I ) has been shown’ to be equivalent to the geometrical expression for weight. To see this consider one dimension of velocity or momentum space, then each w will refer to a velocity or momentum range including the corresponding n molecules, which we can refer to as s. If we use the approximation: log K! = N log
K
-
s
(2)
then the statistical part of ( I ) is readily seen to be equivalent to (nlnlnznz . . . .nk”k) so that, ( I ) may be written: XX
which refers to one dimension say of velocity space. S o w it is noteworthy in connection with this theory that the w’s (and hence the s’s) are arbitrary, (except that in the quantum theory they determine energy levels and are integral multiples of h) and it is in connection with this fact that the geometrical treatment differs from the statistical one, for the prevailing mode is to choose the w’s as all equal (or at least constant), and then one can readily employ the familiar method used by Boltzmannof maximizing W thru variation of the n’s, keeping the total energy constant, and thereby derive the Maxwell Boltzmann distribution law. I t is equally permissible, however, to keep the n’s constant and let the w’s which are the geometrical part of ( I ) vary, instead of Boltzmann’s method of keeping thew’s constant andvarying the n’s, i.e. the statistical part. And in particular we may take each n equal to unity, thereby removing the statistical aspect entirely and then if we refer to (3) instead of ( I ) , the s’s which correspond to the w’s will be the range2 in velocity space corresponding to separate molecules: I
ri=-- Ui+! - ui Nf(Ui) 2
Kimball: J. Phys. Chem., 33, 1558 (1929). Kimball: loc. cit.
(4)
PRINCIPLES O F LEAST WORK AND MAXIMCM PROBABILITY
613
And these r’s are the velocity differences between successive molecules in velocity space used in deriving the geometrical expression for weight :
W
=
NN(rlrz .
. . . rN)
(5)
Thus it is readily seen that (3) reduces to ( j ) when we take s, = ri and nl = n2 = n3 . . . . nk = I . The expression ( j ) is geometrical because the variable, vital part is the geometrical velocity (or action) range that includes each particle. On the other hand, the variable, vital part of ( I ) and 13) is statistical being the n’s, necessarily integers associated with arbitrary constant compartments of p-space.
2. Entropy, Strain, Weight, A-Priori Probability and the Third Law of Thermodynamics If we interpret the r’s given by (4)as momentum ranges instead of velocity ranges, then the weight that measures the thermodynamic probability, taking account of ordinary space ranges as well as three dimensions in momentum space, is given by: \I- = N3’(rlr2 . , . . rh.)31-N ( x r ) 3 h ’ ~ ” (6) where the right member gives the weight in the terms of the range r of a molecule in its mean energy state corresponding to the temperature in question. (See eqs. ( 3 2 ) and (39) of Entropy and Probability).’ When (6) is substituted in Boltzmann’s equation we have the known expression for entropy of monatomic gas, S = klog 11- C = R logT’(2~emkT)~’~C = RlogVW 3kL:logri C ( 7 )
+
+
and hence :
+
dV dri dT’ dS=Ry+3kL:-=R-+3RT. ri v
+
dr r
Equation (8) shows that the change in Entropy equals k times the sum of the corresponding strains for the separate molecules, both in ordinary space and in momentum space. Entropy thus appears as an extensive physical quantity, being k times the integral of the strains plus a constant, i.e., it is k times the total strain, and the constant of integration gives the lower boundary from which strain is to be measured. Thus if we let Y represent the total strain or yielding of the gas, we have:
S
=
kY; Y = log 1%’; W
=
ey
(9)
The concept weight is introduced into physics to avoid the indeterminate aspects of thermodynamic probability which should be a proper fraction, whereas the denominator to be used is unknown. Eq. (9) shows how this weight’ that measures the a-priori probability of a gaseous state is related to the strain. This gives additional physical significance to the idea of weight, heretofore known mainly as the volume in so called y-space, and perhaps pushes thermodynamic probability further into the background. We show Kimball: LOC.cit.
614
W. S . KIMBALL
below how the state of maximum probability corresponds to the equilibrium equations between stress and strain, and represents by (9) a state of maximum strain. It is like the statics of indeterminate frames (see below) where the equilibrium equations correspond to a state of maximum strain for given internal energy. That is to say, the gas yields as much as it can, subject to the boundary conditions on its volume and energy. hltho these results are derived from considerations of monatomic gas obeying the gas law, they are readily seen to apply to complex gases having more than three degrees of freedom, since these gases also have their entropies expressible in terms of momentum and action range, according to Entropy and Probability,l equation (48), which form gives the dS like (8) as the sum of the strains. The present treatment suggests that perhaps any probability theory available to calculate entropy according to Boltzmann’s equation (6) (or statistical theory resting on probability) has value by virtue of its relation to the forces and balancing strains, for mechanical strain appears to be the corner stone of a-priori probability wherever (9) can be applied to gases and other branches of physics. It is to be emphasized that 19) refers oiily to a-priori probability. It is clear that there are other types of probability which are not based on the probability axioms and which are not related to entropy and mechanical strain according to (9). Thus the usual probability expression associated with the Maxwell distribution law, d S , ’ S = f(uj du, is du times the reciprocal of the weight per molecule, w = S r = I f!u). I t is this latter (Type h probability) which is based on the probability axioms and is related to entropy and strain according to (9). On the other hand f(,u) du can not be used in ( 7 ) or (9) to give entropy and strain, and thus represents a type of probability (Type C ) to which Roltamann’s equation does not apply. (See Entropy and Probability,’ 1,eqs. ( 1 5 ) and ( 2 5 ) ) . When entropy is clearly related to probability according t o Boltzmann’s relation, no case appears where it has been shown that these are not related to mechanical strain according to (9). The present paper shows how (9) applies in ordinary gas theory, and likewise it is planned to show how it applies to fluctuations in gas where equilibrium prevails. The applicability of (9) to electronics, the Schott effect and other cases remains t o be proved ordisproved. If (9) holds true all the way down t o absolute zero, the corresponding interpretation of the third law of thermodynamics is that the total strain of crystalline substances at the absolute zero is zero. If entropy is proportional to strain, it suggests that the equilibrium state of maximum entropy and strain is brought about thru the action of stresses. To prove this we note how forces may act in velocity and momentum space, first, however taking note how the principle of maximum ent’ropy,probability and strain is related to the principle of minimum internal energy.
’ Iiimball: Loc. cit.
PRINCIPLES OF LEAST WORK AND MAXIMEM PROBABILITY
615
3. The Principle of Maximum Entropy, Probability and Strain is Equivalent
to the Principle of Least Work or Internal Energy It is well known that equilibrium conditions in gas theory correspond to the state of maximum entropy and probability. If we restrict attention to one dimension of velocity space, the probability is given by ( 5 ) above. The hlaxwell distribution law of velocities is obtained by maximizing the probability ( 5 ) subject to the condition that the energy remain constant. Csing the Lagrange method of multipliers we form the function:
F
=
W
+ XU
(10)
where U the energy is a homogeneous quadratic function of the velocity coordinates, and take the partial derivatives of ( I O ) .
-(log f(ui)) + Xmui
aF - _ - -Wd dui
dui
=
Rd -(log dui
ri)
+ Xmui=
o
(11)
Equations (11) are of the same form as if W mere kept constant and U were being minimized. Thus we see that these equations (11), and the Maxwell distribution law arising from them, represent the state of minimum internal energy for a given constant probability (or entropy) as well as the state of maximum probability for a given constant internal energy. Altho (11) are merely conditions for the extrema1 values of W or U subject to the constancy of the other, it, is readily shown by taking the second partials of U subject to the constant W condition that these partial derivatives are positive and satisfy the sufficiency condition to make U a minimum rather than a maximum. The equivalence of these two principles also holds in the domain of elasticity theory as applied to engineering problems. Thus the problem used in Church’s “Mechanics of Internal Work”’ to illustrate Castigliano’s Principle of Least Work, may be treeted by the Lagrange method for conditional maxima and minima. The extension of a bar in the frame is given in terms of the tension by: y = C T ; and C’ = L:’EA (12) and L is the length, E the elastic modulus, and .1 the cross section of the ba and the internal energy is given by:
U
=
+
~(CITI’ . . . .
+ C6T6‘)
(13)
Then if P is the load, we have from statics: (see figure1)
where Q, R and S are convenient notations for the indicated zero value I. P. Church: ‘‘Mechanics of Internal Work.”
+X +0+ +0 +p CzTz + 0 + 0 + 0 - p 0
-_ - C1Tl aF a rl -_ aF aT*
=
Y
/\
0
9/ /
=
piT\ \
a F - C 3 T 3 + -X+ p + 0 + O + O = 0 -dTa 4;
aF -_ - C1T4 dT4
+ 0 + 0 + -+ + 0 = 0
(I6)
6
2
\j 2
\ \
/i
/ :I
I
\3
\/ P
/ 6
2
PRINCIPLES O F LEAST WORK AND MAXIMUM PROBABILITY
617
may take place by collisions without alteration of its position in ordinary space. Thus a molecule of gas in equilibrium or an atom of a solid may keep its position of equilibrium in ordinary space while changing its position in velocity space during an interval of time dt. Such a time rate of change of momentum involves, according to Newton’s law, the operation of a force. This force which in the absence of displacements does no resultant work in ordinary space, may be thought of as doing work in velocity space or momentum space since the molecule on which it acts takes on a corresponding change of energy as the force acts to change its position in velocity or momentum space. Thus forces must (to the extent that Kewton’s laws apply to agitated molecules) be thought of as operating not only in ordinary space but in velocity and momentum space. The corresponding energy equations take the form :
and W - W o =
11,.
mudu.
where the limits x and x, refer to the same place in ordinary space a t different times and to different places in momentum space where momenta are mu and muo respectively. This means that the interval x-x, over which the integrand is summed, amounts to zero when plus and minus signs are considered, altho physically it is ordinary space thru which the force acts (back and forth. 5 . Theorem: The Equations that Determine the State of Maximum Entropy
or Probability represent Equilibrium between Stresses and Strains in Velocity or Momentum Space I n this section we show that equations (11) or (16) which determine the state of maximum probability or least internal work, are to be interpreted as equilibrium equations between stress and strain. To do this, we replace X, the undetermined constant, by its value -W ’kT which is determined from the distribution function in the usual way. Then if we transpose and multiply both sides of
( I I)
du by k T -and drop the subscripts, we obtain: dx
We here view dx = u dt as the absolute value of distance travelled (of plus or minus sign) in time dt, whose resultant is taken to be zero, and the du is the change in velocity magnitude, Le. the change in agitation velocity according to the last section, so that the force referred t o acts only in velocity
618
W. S. KIMBALL
or momentum space. Where molecules are free to move it is always real and finite (zero when the above du is zero); and distinct from the force exerted by the molecules in ordinary space, (this latter will be zero when the plus and minus values of du,Jdt cancel each other). Equation (20) refers to the individual molecule in the Maxwell distribution where the velocity is u, and if the differential coefficients were known, the magnitude of (20) could be determined. Since ( 2 0 ) is like ( I I ) an equilibrium equation, and since the right member is the force corresponding to one molecule, the left member must be the reacting force per molecule, which is seen to include the corresponding strain multiplied by the elastic coefficient per molecule. Thus dr/r is a true strain, Le., a dimensionless physical quantity, a change in range per unit range. This range is the distance that separates successive molecules in velocity space, (or the excess distance in ordinary space acquired between them in unit time). The way in which the foregoing fits into the theory of elasticity becomes more clear if we re-call that stress or internal force per unit area may be thought of as internal energy per unit volume. Thus in electrostatics where the field strength is F, the energy per unit volume is given by F2/z and this is also the expression for tension or ether stress along lines of force. Also for the Pascal case of hydrostatic equilibrium, the pressure p is the potential energy per unit volume, and also represents the three normal stresses. Consider also the isot,hermal bulk modulus of elasticity p, the ratio between pressure differences and the corresponding change in volume per unit volume:
Here the dp is a force per unit area that refers not to the Pascal hydrostatic force equal in all directions, but to that stress which is balanced by the strain according to (zI), being the change in potential energy per unit volume. For the Maxwell distribution of velocities we have from (20): muldul - mu?dus --muxdux kT = -- __ - . . . . drIIr, drzir, drs,r y
(22)
which shows that for gas in equilibrium at temperature T, there is a constant ratio, Le., an elastic coefficient per molecule, k T which is the ratio between energy difference and corresponding strain for the various positions of a molecule throughout velocity space. As the denominators of ( 2 2 ) are true strains, so it is clear that the numerators are pressure differences due to energy changes, which tend to change the positions in velocity space of molecules on which they act, being balanced by the corresponding strains. Each position ui in velocity space is [under equilibrium conditions) maintained by some molecule (or other) under action solely of pressure due to impacts, and this must be an unbalanced force as between molecules of different energy and position in velocity space. Thus the equations ( 1 1 ) which correspond to the Maxwell distribution are seen to be equilibrium
PRINCIPLES O F LEAST WORK AND MAXIMEX PROBABILITY
619
equations between the stress and strain in velocity and momentum space. Accordingly this distribution seems to be the result of forces acting in velocity space. Likewise it will readily be understood that equations (16) can he shown to be the equilibrium equations between stress and strain. (Express the undetermined Greek constants in terms of the y’s with the help of ( 1 2 ) and divide by the C’s). 6 . Magnitude of Stress and Strain acting in Velocity Space, where Thermal
Equilibrium prevails, in Gas obeying the Gas Law The proportion ( 2 2 ) still holds if we sum the numerators and sum the denominators for the n molecules in unit volume:
or multiplying by n:
where the numerator is seen to be the total force for all n molecules per C.C. due t o energy differences of molecules that react against each other, and the modulus of elasticity p is the familiar isothermal bulk modulus for gas obeying the gas law according to (21). The numerator f, is seen to be like the usual p = +(nmv2)= nmu?, except that 2mu of the latter, the change in momentum per impact in the x direction, is replaced by zmdu, the excess change of momentum that has to be balanced by strain in momentum space. To calculate the magnitude of the stress f, above, we note that the differentials in (23) and ( 2 4 ) ’ which generally above have their usual interpretation as variables approaching zero, must be given a special interpretation if the numerators of these equations are to represent the actual stress or total unbalanced pressure due to energy differences. Thus if we take in the manner of elementary kinetic theory, ui !z as the number of impacts per second and zmdui as the unbalanced change in momentum per impact for this component of this molecule, the product gives the excess change in momentum per second or unbalanced pressure as between two molecules at ui in velocity space. But the ri used throughout this paper is precisely the expression for velocity (or momentum) differences between successive molecules in velocity space, and it is only when such differences are used in (23) and ( 2 4 ) that the energy differences shown there represent unbalanced pressure of molecules that jostle each other in equilibrium. The same result is reached if no limit is imposed on the arbitrariness of the differentials which might be zero but then the number of molecules would increase indefinitely and still satisfy the distribution law. Thus for the stress or force per unit area in the x direction and acting only in velocity space:
W. S. KIMBALL
620
j2nemkT)t = f i e kT n where the momentum range mr for the molecule in its mean energy state is introduced with the help of (4) and mu2 = k T for one component of the root mean square velocity. Also see the writer’s Entropy and Probability’, equation (38). And likewise the corresponding strain is:
f, = nmu du = nmur = nu
dr - zu du r a2
- _ - E
zu(znemkT)* = &e ma2n
n
I t is noteworthy that no change of T occurs with the differentials of these equations because we are copsidering changes within gas where temperature equilibrium prevails, and also that the strain is a pure number depending only directly on the density and affected by temperature changes only indirectly according to the gas law. It is t o be emphasized that the stress ( 2 5 ) does not in general operate to change momentum in ordinary space except as applied to motions where the vector sum of the changes in ordinary space is zero, although they add up to increase agitation energy and absolute velocity, thus corresponding to a shift of position in velocity and momentum space. And yet f, is a true stress in that it acts on unit area like hydrostatic pressure and is an energy difference for the n molecules per unit volume of ordinary space and would cause a time rate of change of momentum (in momentum space) unless balanced as per (24) and (11). Furthermore, it is an isothermal stress like the dp of ( 2 r j with the same modulus of elasticity p = nkT; and again if we divide both members of ( z I j by n and interpret dpln as the pressure difference per molecule, then the elastic modulus per molecule will be k T for ( 2 1 ) as it is in ( 2 2 ) . If there were uniform molecular distribution in velocity space, Le., if there were equal intervals between successive molecules in velocity space, then dri would be zero and there would be no reacting force (kT times the strain) to counterbalance the forces represented by the numerators of ( z z ) , ( 2 3 ) and (24). And those forces would operate by elementary mechanics to increase the energy of the slow molecules and reduce the energy of the fast moving ones, thus reducing the intervals ri by bringing the molecules together in velocity space. But a smaller ri makes an increased strain (other things being equal) and hence increased reacting force. The condition of equilibrium is thus a stress given by the numerators of ( 2 3 ) and (24) acting in velocity space through forces shown as numerators of (23) that tend to reduce the differences between agitation velocities (to reduce intervals in velocity space) , which forces are balanced by the increased strains. Comparison of ( 2 1 ) and (24) shows that molecules “object” to all occupying the same position in velocity space with the same vigor (same modulus of elasticity) that they object to occupying it in ordinary space, and apparently for the same reason, Le., more of them together in unit interval causes bigger reaction. LOC.cit.
PRINCIPLES OF LEAST WORK AND MAXIMCM PROBABILITY
6 2I
7. The Second Law of Thermodynamics According to this law automatic changes within a closed system always involve an increase of entropy till the equilibrium state of maximum entropy and probability is reached subject to the boundary conditions on the volume and energy of the system. No ergodic hypothesis is needed, according to the present treatment, to account for this fact that the rate of change of entropy is always positive: d - s > O
dt
For we have seen that the state of maximum entropy corresponds to the equilibrium equations between stress and strain in ordinary and momentum space. And likewise the unstable state corresponds to a failure to satisfy the equilibrium equations between stress and strain within the gas. That is to say the stresses within the gas are not completely balanced in the absence of the equilibrium condition of maximum entropy. And this refers to stresses acting in momentum and velocity space as well as ordinary space according to 4 and 5 . Thus a gas at uniform temperature and pressure whose molecules were uniformly spaced, but wherein a Maxwellian distribution of velocities did not prevail, would involve stresses acting in velocity space which as pointed out in 6, would not be balanced by the strains corresponding to the equilibrium condition. Hence an immediate rearrangement in velocity space would take place involving a time rate of change of momentum of most of the molecules under the action of internalforces, i.e. : stresses would act in velocity and momentum space. A simple illustration of the second law is found by considering equal quantities (X molecules) of the same type of gas at two different temperatures TI and Tz but a t the same pressure and separated by any conceivable barrier impervious to heat conduction. If we take T1 > Tz the first gas will occupy more volume than the second (Le. V1>VQ) according to the gas law (since there are equal numbers of molecules). Hydrostatic equilibrium prevails since-the pressure is taken to be uniform, and we assume that two gases are thermally insulated from the outside world by fixed walls impervious to heat tho separated from each other by a removable non conducting barrier. Now remove the imaginary barrier in such a way as not to disturb the two gases by the removal process, and note the operation of the second law of thermodynamics in this container where hydrostatic equilibrium prevails (but not thermodynamic). Molecules of the high temperature gas, having more kinetic energy, will as they impact with those of the cooler gas impart by elementary mechanics some of their kinetic energy to the latter until at length temperature equilibrium prevails a t T = (T1+T2)/z,while the total agitation energy of the two gases together remains constant. This increase of energy of the cooler molecules involves time rate of change of momentum and the action of stresses (forces) within the gas to cause that rate of change. And the corresponding increased total strain parallels the known increase
622
W. S . KIMBALL
of entropy according to (7), ( 8 ) , and (9). And thruout this change no work is done in ordinary space wherein hydrostatic equilibrium is maintained. Although no attempt is here made to extend this explanation to the multitudinous complicated cases to which the second law applies, yet it would seem that it would apply and account for the changes of states involved in the workings of the second law in all cases where the relation between entropy and strain is correctly given by (9).
8. Experimental Verification We note that these relations between entropy, strain and a-priori probability are an immediate consequence of using in Boltzmann’s equation the geometrical expression for weight (6) rather than the familiar statistical one. I t was pointed out, however, in the writer’s Entropy and Probability’ that these expressions are mathematically equivalent and interchangeable. This makes the problem of distinguishing experimentally between the two points of view a difficult one. For if one starts with two equivalent expressions for entropy, one obtains equivalent theoretical results whenever equivalent mathematical lines of reasoning are applied to the two treatments. The Maxwell distribution of velocities viewed as the most probable distribution may be thought of as brought about spontaneously and explicable in statistics according to the ergodic hypothesis. The present theory indicates however, like Boltzmann’s H-theorem that it comes about as a result of forces. The question arises, are the many so called “spontaneous” deuzatzons from equilibrium which involve corresponding changes of entropy also explicable as due to the action of forces? The reasoning of the present paper indicates an affirmative answer for such probability deviations prouzded their probability is of the a-priori Type A and measured according to the probability axioms and related to entropy and strain according to (9) by Boltzmann’s relation. There seems to be no case where it has been positively shown that there are no veiled forces to account for “spontaneous” changes in entropy like the veiled forces that the present treatment shows are involved in the Maxwell velocity distribution of maximum probability. The detailed application of this method to fluctuations within gas where a Maxwell distribution of velocities prevails, is planned in a subsequent paper. On the other hand the present theory has nothing to say about fluctuations and changes whose probability is not of Type A and hence not related to Entropy according to (9) by Boltzmann’s relation. (See section two above). The most outstanding verification of the present theory, however, seems to be the second law of thermodynamics itself. The ergodic hypothesis, as an explanation of why gas will go from an unstable, improbable state to the equilibrium state of maximum probability, does not appear satisfactory to most physicists and still presents, according to Tolman’, “a baffling problem for further study”. This marks a sore spot and partial failure in the statistical theory. Kimball: loc. cit.
* Tolmsn: “Statistics1
Mechanics”,
21,
39.
PRIXCIPLES OF LEAST WORK AND MAXIMUM PROBABILITY
623
On the other hand, forces certainly do act in velocity and momentum space, and, as pointed out herein, may do work in these realms when no work is done in ordinary space, and cause strains there distinct from any strains in ordinary space. Thus, when entropy is viewed as k times the sum of the strains, it is readily suggested that it is the action of these forces that cause the state of maximum entropy and probability to be reached. This mechanical explanation of the second law is calculated to appeal to those who welcome the modern tendency to extend further the application of mechanical principles, as for example to the action of light photons. The writer is specially indebted to Prof. G. E. Uhlenbeck of the University of Michigan for valuable criticism and discussions. Michigan State College, East Laming, Michigan
