Want more? Advanced embedding details, examples, and help! The course is being developed by an interuniversity group, of which Charles Kittel is chairman Includes bibliographies v. Mechanics, by C. Kittel, W. Knight, and M. Electricity and magnetism, by E. Waves, by F. Crawford, Jr. Advances in Imaging and Electron Physics: Volume The topics reviewed within the 'Advances' sequence disguise a vast variety of issues together with microscopy, electromagnetic fields and picture coding.
This publication is key studying for electric engineers, utilized mathematicians and robotics specialists. During this paintings, the utmost entropy procedure is used to resolve the extension challenge linked to a positive-definite functionality, or distribution, outlined on an period of the true line. The time average value of n at any time t, taken over the time interval r, can then be denoted by [n t ]Tand is defined by.
If we omit explicit indication of the time interval t to be considered, n f implies an average over some appropriately chosen time interval t of appreciable length. In the equilibrium situation of our gas, n tends to be constant and equal to N. Consider an isolated gas consisting of a large number N of molecules. Such situations can occur in two different ways which we shall discuss in turn. Suppose that such a large spontaneous fluctuation of A n has oc curred, e.
What can we then say about the probable behavior of n as time goes on? It is then most likely that the value ni occurs as a result of a fluctuation which is rep resented by a peak whose maximum is near ni as indicated by the peak marked X in Fig.
The reason is the following: It might be possible for a value as large as to occur also as a result of a fluctua tion represented by a peak whose maximum is larger than rti such as the peak marked Y in Fig. The general behavior of n as a function of time is then, however, apparent from Fig. Not only does such a movie strip occur very rarely but, if it does occur, it. Schem atic illustration showing rare instances where the num ber n of molecules in one half o f a box exhibits large fluctuations about its equilibrium value -JiV.
In more physical terms, the value ni corresponds to a very nonuniform distri bution of the molecules and the molecules would have to move in a very special way in order to preserve such nonuniformity. The inces sant motion of the molecules thus almost always results in mixing them up so thoroughly that they become distributed over the entire box in the most random or uniform possible way.
See Figs. Remarks Note that the statements of the preceding paragraph are equally applicable whether the large fluctuation n i N is positive or negative.
If it is positive, the value ni will almost always correspond to the max imum of a fluctuation in n such as that indicated by the peak X in Fig. If it is negative, it will almost always corre spond to the minimum of a fluctuation in n. The argument leading to the conclu sion of the paragraph remains, however, essentially identical. Note also that the statements of the paragraph remain equally valid irrespec tive of whether the change in time is in the forward or backward direction i.
The piston in a is moved to the position b so as to compress the gas into the left half of the box. W hen the piston is sud denly restored to its initial position, as shown in c , the molecules immediately afterward are all located in the left half of the box, while the right half is empty. We use the qualifying word almost since, instead of corresponding to a maximum of a peak such as X, the value ni may very rarely lie on the rising side of a peak such as Y.
In this case n would initially increase, i. Specially prepared initial situations Although a nonrandom situation, where n is appreciably different from iN , may occur as a result of a spontaneous fluctuation of the gas in equilibrium, such a large fluctuation occurs so rarely that it would al most never be observed in practice. Nonrandom situations thus occur quite commonly, not as a result of spon taneous fluctuations of a system in equilibrium, but as a result of interactions which affected the system at some not too distant time in the past.
Indeed, it is quite easy to bring about a nonrandom situa tion of a system by means of external intervention. Examples When a wall of a box is made movable, it becomes a piston. One can use such a piston as shown in Fig. When the piston is suddenly restored to its initial position, all the molecules im mediately afterward are still in the left half of the box. Thus one has produced an extremely nonuniform distribution of the molecules in the box.
Equivalently, consider a box divided into two equal parts by a partition see Fig. If the gas is in equilibrium under these circumstances, the distribution of its molecules is essentially uniform throughout the left half of the box.
Im agine that the partition is now suddenly removed. The molecules immediately afterward are then still uniformly distrib uted throughout the left half of the box. This distribution, however, is highly nonuniform under the new conditions which leave the molecules free to move through out the entire box. Suppose that an isolated system is known to be in a highly nonran dom situation, e. Then it is essentially irrelevant whether the system got into this condition by virtue of a very rare spontaneous fluctuation in equilibrium, or whether it got there by virtue of some form of prior external intervention.
Irrespective of its past history, the subsequent behavior of the system in time will thus be similar to that discussed previously when we considered the decay of a large fluctuation in equilibrium. In short, since nearly all possible ways in which the mole cules of the system can move will result in a more random distribution of these molecules, the situation of the system will almost always tend to change in time so as to become as random as possible. After the most random condition has been attained, it exhibits then no further tendency to change, i.
For example, Fig. The important conclusion which we have reached in this section can thus be summarized as follows: If an isolated system is in an appreciably nonrandom sit uation, it will except for fluctuations which are unlikely to be large change in time so as to approach ultimately its most random situation where it is in equilibrium.
Note that the preceding statement does not make any assertions about the relaxation time, i. The actual magnitude of this time depends sensitively on details of the system under consideration; it might be of the order of microseconds or of the order of centuries. A movie played backward would show the pictures in the re verse order c , b , a. Example Referring to Fig. The left half of the box contains N molecules of gas, while its right half is empty.
Imagine now that the partition is suddenly removedbut only partly as shown in Fig. But the time required until the final equilibrium condition is reached will be longer in the experiment of Fig. The relaxation tim e is indicated by Tr. Irreversibility The statement 7 asserts that, when an isolated macroscopic system changes in time, it tends to do so in a very definite directionnamely, from a less random to a more random situation.
We could observe the process of change by taking a movie of the system. Suppose now that we played the movie backward through a projector i. We would then observe on the screen the time-reversed process, i. The movie on the screen would look very peculiar indeed since it would portray a process in which the system changes from a more random to a much less random situationsomething which one would almost never observe in actuality.
Just by watching the movie on the screen, we could conclude with almost complete certainty that the movie is being played backward through the projector. Example For example, suppose one filmed the process that takes place after the parti tion in Fig. The movie played forward through the projector would show the gas spreading out as indicated in Fig. Such a proc ess is quite familiar. On the other hand, the movie played backward would show the gas, initially distributed uniformly throughout the box, concentrating itself spontaneously in the left half of the box so as to leave the right half empty.
Such a process is virtually never observed in actuality. This does not mean that this process is impossible, only that it is ex ceedingly improbable. What the back ward movie shows could happen in actu ality if all the molecules moved in an extremely special w ay. This cartoon is humorous because it portrays the reverse of an irreversible proc ess.
The indicated sequence o f events c o u l d happen, but it is exceedingly unlikely that it ever would. A process is said to be irreversible if the time-reversed process the one which would be observed in a movie played backward is such that it would almost never occur in actuality.
But all macroscopic systems not in equilibrium tend to approach equilibrium, i. Hence we see that all such systems exhibit irreversible behavior. Since in everyday life we are constantly surrounded by systems which are not in equilibrium, it becomes clear why time seems to have an unambiguous direction which allows us to distinguish clearly the past from the future.
Thus we expect people to be bom, grow up, and die. We never see the time-reversed process in principle possible, but fantastically unlikely where someone rises from his grave, grows progressively younger, and disappears into his mothers womb. Note that there is nothing intrinsic in the laws of motion of the par ticles of a system which gives time a preferred direction. Indeed, suppose that one took a movie of the isolated gas in equilibrium, as shown in Fig. Then each molecule would retrace its path in time.
The gas would thus reconcentrate itself in the left half of the box. The preferred direction of time arises only when one deals with an isolated macroscopic system which is somehow known to be in a veiy special nonrandom situation at a specified time fi. If the system has been left undisturbed for a very long time and got into this situa tion as a result of a very rare spontaneous fluctuation in equilibrium, there is indeed nothing special about the direction of time.
As already pointed out in connection with the peak X in Fig. Finally it is worth pointing out that the irreversibility of spontane ously occurring processes is a matter of degree. The irreversibility becomes more pronounced to the extent that the system contains many particles since the occurrence of an orderly situation then becomes increasingly unlikely compared to the occurrence of a random one.
Example Consider a box which contains only a sin gle molecule moving around and colliding elastically with the walls. If one took a movie of this system and then watched its projection on a screen, there would never be any way of telling whether the movie is being played forward or back ward through the projector. Consider now a box which contains N molecules of an ideal gas. Suppose that a movie of this gas is projected on a screen and portrays a process rn which the molecules of the gas, originally dis tributed uniformly throughout the box, all become concentrated in the left half of the box.
What could we conclude? On the average, 1 out of every 16 frames of the film would show all the molecules in the left half of the box.
Hence we can not really tell with appreciable certainty whether the film is being played forward or backward. See Fig. It is much more likely that the film is being played backward and portrays the result of a prior intervention, e.
One could then be almost completely cer tain that the film is being played back ward. Construction o f a possible past history for Fig. The short line segment emanating from each particle in dicates the direction of its velocity in this case. Computer-made pictures showing 4 particles in a box.
T h e resulting evolution of the system in. T he short line segment emanating from each particle indicates the direction of the particles velocity. No velocities are indicated. Computer-made pictures showing 4 0 particles in a box. T h e graphs de scribe Figs. T h e right half o f each graph shows the approach of the system to equilibrium.
The entire domain o f each graph shows the occurrence of a rare fluctuation which might occur in an equilibrium situation. The information presented is otherwise the same as that in Fig. By thinking in detail about the simple case of the ideal gas of N mole cules, we have grappled with all the essential problems involved in understanding systems consisting of very many particles. Indeed, most of the remainder of this book will consist merely of the systematic elaboration and refinement of the ideas which we have already dis cussed.
To begin, let us illustrate the universal applicability of the basic concepts which we have introduced by considering briefly a few further examples of simple macroscopic systems. Id ea l system o f N spins Consider a system of N particles each of which has a spin and an associated magnetic moment of magnitude ju 0. The particles might be electrons, atoms having one unpaired electron, or nuclei such as protons.
The concept of spin must be described in terms of quantum ideas. For the sake of simplicity, we shall designate these two possible orientations as up or down, respec tively.
For the sake of simplicity, we may re gard the particles as essentially fixed in position, as they would be if they were atoms located at the lattice sites of a solid, f We shall call the system of spins ideal if the interaction between the spins is almost negligible.
This is the case if the average distance between the par ticles with spin is so large that the magnetic field produced by one moment at the position of another moment is small enough to be nearly negligible.
This is the usual situation if the particle is negatively charged. In this case the magnetic moment points down when the spin points up, and vice versa. The ideal system of N spins has been described entirely in terms of quantum mechanics, but is otherwise completely analogous to the ideal gas of N molecules. In the case of the gas, each molecule moves about and collides occasionally with other molecules; hence it is found some times in the left and sometimes in the right half of the box.
In the case of the system of spins, each magnetic moment interacts slightly with the other magnetic moments, so that its orientation occasionally changes; hence each magnetic moment is found to point sometimes up and sometimes down. In the case of the isolated ideal gas in equilib rium, each molecule is equally likely to be found in either the left or right half of the box. Similarly, in the case of the isolated system of spins in equilibrium in the absence of any externally applied magnetic field, each magnetic moment is equally likely to be found pointing either up or down.
We can denote by n the number of spins pointing up, and by n' the number of spins pointing down. Indeed, when N is large, nonrandom situations where n differs appreciably from occur almost always as a result of prior interaction of the isolated system of spins with some other system. D istribution o f energy in an ideal gas Consider again the isolated ideal gas of N molecules. We reached the general conclusion that the time-independent equilibrium situation attained by the system after a sufficiently long time corresponds to the most random distribution of the molecules.
In our previous discussion we focused attention solely on the positions of the molecules. We then saw that the equilibrium of the gas corresponds to the most ran dom distribution of the molecules in space, i. But what can we say about the velocities of the molecules? Here it is useful to recall the fundamental mechanical principle that the total energy E of the gas must remain constant since the gas is an isolated system.
This total energy E is equal to the sum of the energies of the individual gas molecules, since the potential energy of interaction be tween molecules is negligible. Hence the basic question becomes: How is the fixed total energy of the gas distributed over the individual molecules?
It is possible that one group of molecules might have very high energies while another group might have very low energies. The timeindependent equilibrium situation which is ultimately attained corre sponds therefore to the most random distribution of the total energy of the gas over all the molecules.
Each molecule has then, on the average, the same energy and thus also the same speed, t In addition, since there is no preferred direction in space, the most random sit uation of the gas is that where the velocity of each molecule is equally likely to point in any direction.
Pendulum oscillating in a gas Consider a pendulum which is set oscillating in a box containing an ideal gas. If the gas were not present, the pendulum would continue oscillating indefinitely with no change in amplitude. We neglect fric tional effects that might arise at the point of support of the pendulum. But in the presence of the gas, the situation is quite different.
The molecules of the gas collide constantiy with the pendulum bob. In each such collision energy is transferred from the pendulum bob to a molecule, or vice versa.
What is the net effect of these collisions? Again this question can be answered by our familiar general argument without the need to consider the collisions in detail. The energy Ej kinetic plus potential of the pendulum bob plus the total energy Eg of all the gas molecules must remain constant since the total system is isolated if one includes the earth which provides the gravitational attraction.
If the energy of the bob were transferred to the gas mole cules, it could be distributed over these many molecules in many dif ferent ways instead of remaining entirely associated with the bob. A much more random situation of the system would thus result. Since an isolated system tends to approach its most random situation, the pendulum gradually transfers practically all its energy to the gas mole cules and thus oscillates with ever-decreasing amplitude.
This is again a typical irreversible process. After the final equilibrium situation has been reached, the pendulum hangs vertically, except for very small oscillations about this position. The energy of any one molecule fluctuates quite appreciably in the course of time as a result of its collisions with other molecules.
But when each molecule is observed over a suffi ciently long time interval t , its average energy over that time interval is the same as that of any other molecule. J A detailed analysis would argue that the pendulum bob suffers more collisions per unit time with the molecules located on the side toward which the bob moves than with the molecules located on the other side. As a result, collisions in which the bob loses energy to a molecule are more frequent than collisions in which it gains energy from a molecule.
The pendulum is shown successively a soon after it is set into oscillation, b a short time thereafter, and c after a very long time. A movie played backward would show the pic tures in the reverse order c , b , a.
Note one further point of interest. In the initial nonrandom situa tion, where the pendulum bob has a large amount of energy associated with it, this energy can be exploited to do useful work on a macro scopic scale. For example, the pendulum bob could be made to hit a nail so as to drive it some distance into a piece of wood.
After the final equilibrium has been reached, the energy of the pendulum bob has not been lost; it has merely become redistributed over the many molecules of the gas. But there is now no easy way to use this energy to do the work necessary to drive the nail into the wood.
Indeed, this would necessitate some method for concentrating energy, randomly distributed over the many gas molecules moving in many directions, so that it can exert a net force in only one particular direction over an appreciable distance.
Arrangement in w hich the pendu lum bob can b e made to strike a nail and thus do work on it by driving it into a piece o f wood. Simplicity o f the equilibrium situation The discussion of the preceding sections shows that the equilibrium situation of a macroscopic system is particularly simple.
The reasons are the following: i The macrostate of a system in equilibrium is time-independent, except for ever-present fluctuations. Quite generally, the macrostate of a system can be described by certain macroscopic parameters, i. For example, the number n of molecules located in the left half of a box of gas is such a macroscopic parameter.
When the system is in equilibrium, the average values of all its macroscopic parameters remain constant in time, although the parameters themselves may ex hibit fluctuations ordinarily quite small about their average values.
The equilibrium situation of a system is, therefore, simpler to treat than the more general nonequilibrium case where some macroscopic parameters of the system tend to change in time. The system in equilibrium is thus characterized in a unique way. In particular, this has the following implications:.
For example, consider an isolated gas of N molecules in a box. These molecules may originally have been confined by a par tition to one half of the box or to one quarter of the box the total energy of the molecules being assumed the same in each case. But after the partition is removed and equilibrium has been attained, the macrostate of the gas is the same in both cases; it corresponds merely to the uniform distribution of all the molecules through the entire box.
For example, consider again the isolated gas of N identical molecules in a box. Suppose that the volume of the box is V, while the constant total energy of all the molecules is E.
If the gas is in equilibrium and thus known to be in its most random situation, then the molecules must be uniformly dis tributed throughout the volume V and must, on the average, share equally the total energy E available to them. If the gas were not in equilibrium, the situation would of course be much more complicated. The distribution of molecules would ordinarily be highly nonuniform and a mere knowledge of the total number N of molecules in the box would thus be completely in sufficient for determining the average number ns of molecules in any given subvolume Vs of the box.
Observability o f fluctuations Consider a macroscopic parameter describing a system consisting of many particles. If the number of particles in the system is large, the relative magnitude of the fluctuations exhibited by the parameter is ordinarily veiy small. Indeed, it is often so small as to be utterly neg ligible compared to the average value of the parameter. As a result, we remain usually unaware of the existence of fluctuations when we are dealing with large macroscopic systems.
On the other hand, the ever-present fluctuations may be readily observed and may become of great practical importance if the macroscopic system under con sideration is fairly small or if our methods of observation are quite sen sitive.
Several examples will serve to illustrate these comments. Density fluctuations in a gas Consider an ideal gas which is in equilib rium and consists of a large number N of molecules confined within a box of vol ume V. Focus attention on the number ns of molecules located within some spec ified subvolume V, inside the box. This number n, fluctuates in time about an average value volume element Vs having linear dimen sions of the order of the wavelength of light. But in the case of our ideal gas, the average number n of molecules in a volume as small as Vs is quite small and fluctuations An, in the number ns of molecules within Vs are no longer negligible compared to ns.
The gas can, therefore, be expected to scatter light to an appreciable extent. Indeed, the fact that the sky does not look black is due to the fact that light from the sun is scattered by the gas mole cules of the atmosphere. The blue color of the sky thus provides visible evidence for the importance of fluctuations.
When Vj is large, the average number ns of molecules is also large. In accordance with our discussion of Sec. Then we would be interested in knowing what happens in a. Fluctuations of a torsion pendulum Consider a thin fiber stretched between two supports or suspended from one support under the influence of gravity and carrying an attached mirror.
When the mirror turns through a small angle, the twisted fiber provides a restoring torque. The mirror is thus capable of performing small angular oscillations and constitutes accordingly a torsion pendu lum. Since the restoring torque of a thin fiber can be made very small and since a beam of light reflected from the mirror provides a very effective way of detecting small angular deflections of the mirror, a torsion fiber is commonly used for very sensitive measurements of small torques.
For example, it may be recalled that a torsion pendulum was used by Cav endish to measure the universal constant of gravitation and by Coulomb to measure the electrostatic force between charged bodies.
When a sensitive torsion pendulum is in equilibrium, its mirror is not perfectly still, but can be seen to perform erratic angular oscillations about its average equilibrium orientation.
The situation is analogous to that of the ordinary pen dulum, discussed in Sec. These fluc tuations are ordinarily caused by the random impacts of the surrounding air molecules on the mirror. Torsion pendulum formed by a m ir ror mounted on a thin fiber. In that case the total energy of the torsion pendulum would still consist of two parts, the en ergy due to the angular velocity of the mirror moving as a whole, plus the en ergy Ei due to the internal motion of all the atoms of the mirror and fiber.
The atoms are free to perform small vibra. Such particles are not at rest, but are seen to be constantly moving about in a highly irregular way. This phenomenon is called Brownian motion because it was first observed in the last century by an English botanist named Brown. He did not understand the origin of the phenomenon. It remained for Ein stein in to give the correct explana tion in terms of random fluctuations to be expected in equilibrium. A solid par ticle is subject to a fluctuating net force due to the many random collisions of the particle with the molecules of the liquid.
Since the particle is small, the number of molecules with which it collides per unit time is relatively small and accord ingly fluctuates appreciably. In addition, the mass of the particle is so small that any collision has a noticeable effect on the particle. The resulting random mo tion of the particle thus becomes large enough to be observable. Brownian motion o f a solid particle, cm in diameter, suspended in w ater and observed through a m icro scope.
The three-dimensional motion of such a particle, as seen projected in the horizontal plane o f the field of view of the microscope, is shown b y this diagram where the lines join con secutive positions of the particle as observed at second inter vals.
Perrin, Atoms, p. Van Nostrand Company, In c. Voltage fluctuations across a resistor If an electrical resistor is connected across the input terminals of a sensitive electronic amplifier, the output of the amplifier is observed to exhibit random voltage fluctuations. Neglecting noise originating in the amplifier itself, the basic reason is the random Brownian mo tion of the electrons in the resistor.
Sup pose, for example, that this random motion leads to a fluctuation where the number of electrons is greater in one half of the resistor than in the other. The resulting charge difference then leads to an electric field in the resistor, and hence to a potential difference across its ends. Variations in this potential difference thus give rise to the voltage fluctuations amplified by the electronic instrument.
A resistor R connected to the input terminals o f a sensitive amplifier whose output is displayed on an oscilloscope. The existence of fluctuations can have important practical conse quences. This is particularly true whenever one is interested in measuring small effects or signals, since these may be obscured by the intrinsic fluctuations ever-present in the measuring instrument.
These fluctuations are then said to constitute noise since their presence makes measurements difficult. For example, it is difficult to use a torsion fiber to measure a torque which is so small that the angular deflection produced by it is less than the magnitude of the intrinsic fluctuations in angular position displayed by the mirror.
Similarly, in the case of the resistor connected to the amplifier, it is difficult to measure an applied voltage across this resistor if this voltage is smaller than the magnitude of the intrinsic voltage fluctuations always present across the resistor, f. Macroscopic systems which are not isolated can interact and thus exchange energy. One obvious way in which this can happen is if one system does macroscopically recognizable work upon some other sys tem. For example, in Fig. Similarly, in Fig.
When the piston is released and moves through some macroscopic distance, the force exerted by A' performs some workf on the system A. It is, however, quite possible that two macroscopic systems can in teract under circumstances where no macroscopic work is done. This type of interaction, which we shall call thermal interaction, occurs because energy can be transferred from one system to the other sys tem on an atomic scale.
The energy thus transferred is called heat. Imagine that the piston in Fig. In this case one system cannot do macroscopic work upon the other, irrespective of the net force exerted on the piston. On the other hand, the atoms of system A do interact or collide with each other almost constantly and-thus exchange energy among themselves. By averaging the measurement over a sufficiently long time, one may, however, dis criminate against the random fluctuations in favor of the applied signal which does not fluctuate in time.
J The term work is used here in its usual sense familiar from mechanics and is thus defined basically as a force multiplied by the distance through which it acts. If each molecule of the gas consists of more than one atom, different molecules can exchange energy by colliding with each other; furthermore, the energy of a single molecule can also be exchanged between its constituent atoms as a result of interaction between them.
Actual photograph o f the noise out put voltage displayed on an oscilloscope in the experimental arrangement of Fig. The compressed spring A ' does work on the gas A when the piston moves through some net macroscopic distance. The compressed gas A ' does work on the gas A when the piston moves through some net macroscopic distance. Schem atic diagram showing two general systems A and A ' in therm al contact with each other.
Similarly, the atoms of system A' exchange energy among themselves. At the piston, which forms the boundary between the gases A and A', the atoms of A interact with the atoms of the piston, which then inter act among themselves and then in turn interact with the atoms of A'. Hence energy can be transferred from A to A' or from A' to A as a result of many successive interactions between the atoms of these systems.
Consider then any two systems A and A' in thermal interaction with each other. For example, the two systems might be the two gases A and A' just discussed; or A might be a copper block immersed in a system A' consisting of a container filled with water.
Let us denote by E the energy of system A i. The question arises, however, how this total energy is actually dis tributed between the systems A and A'. In particular, suppose that the systems A and A' are in equilibrium with each other, i. Let us first discuss the simple situation where the systems A and A' are two ideal gases consisting of molecules of the same kind. Each molecule of A and A' should then have the same average energy. In particular, the average energy e of a molecule of gas A should be the same as the average energy l!
Suppose that the gases A and A' are initially separated from each other and separately in equilibrium. Imagine now that the systems A and A' are placed in contact with each other so that they are free to exchange energy by thermal interaction.
Instead, the sys tems A and A' will exchange energy until they ultimately attain the equilibrium situation corresponding to the most random distribution of energy, that where the average energy per molecule is the same in both systems.
Two gases A and A ', consisting of molecules of the same kind, are initially sepa rated from each other in a. They are then brought into thermal contact with each other in b and are allowed to exchange heat until they reach equilibrium. T h e energy of a gas is denoted b y E, the average energy per mole cule by e. In the interaction process leading to the final equilibrium situation the system with the smaller average initial energy per molecule thus gains energy,while the system with thelarger average initial energy per molecule losesenergy.
The quantity Q is called the h eat absorbed by A in the interaction process and is defined as the increase in energy of A resulting from the thermal interaction process. A similar definition holds for the heat Q' absorbed by A'.
Indeed, in the thermal interaction between two systems, one loses energy while the other gains energy; i. By definition, the system which gains energy by absorbing a positive amount of heat is said to be the colder system; on the other hand, the system which loses energy by absorbing a negative amount of heat i. Thus the systems remain in equilibrium and there occurs no net transfer of energy or heat be tween the systems.
Temperature Consider now the general case of thermal interaction between two systems A and A'. These systems might be different gases whose mole cules have different masses or consist of different kinds of atoms; one or both of the systems might also be liquids or solids.
Qualitatively, most of our preceding comments con cerning the case of two similar gases ought, however, still to be appli cable. What we should then expect and shall explicitly show later on is the following: Each system, such as A, is characterized by a parame ter T conventionally called its absolute temperature which is related to the average energy per atom in the system.
It is not possible to define the concept of absolute temperature more precisely until we have given a more precise definition of the concept of randomness applicable to the, distribution of energy over unlike atoms.
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