data – Idealisation modality
5.2 Mechanical engineering 1 Control Volume Analysis
5.3.3 Statistical mechanics
The statistical mechanics approach is unique to the physics textbooks, and not followed in any of the other textbooks under consideration. For this reason, and because of the detail necessary to communicate the approach in terms of the knowledge modalities, the discussion is fairly dense.
The Second Law of thermodynamics and entropy are introduced in both physics textbooks (Chabay & Sherwood, 2011; Schroeder, 2000) by a look at time reversal invariance of macroscopic processes:
…why does heat flow spontaneously from a hotter object to a cooler object, never the other way? More generally, why do so many thermodynamic processes happen in one direction but never the reverse? This is the Big Question of thermal physics…
(Schroeder, 2000, p. 49)
Macroscopic processes are in general irreversible: ice cubes melt in a cup of hot water, a ball bounces lower off the ground with every bounce and heat energy flows from a hot object to a colder one. These processes take place in accordance with the First Law of thermodynamics, but the reverse processes would not in principle violate the conservation of energy law. As long as the water gets hotter while the ice cube gets colder, the total amount of energy in the ice cube + water system remains conserved. And yet experience shows us that these reverse processes do not take place. The physics textbooks use an approach that “deals with a statistical analysis of microscopic energy that puts limits on what is possible”(Chabay & Sherwood, 2011, p. 472). By contrast, the mechanical engineering text simply states irreversibility as a fact: “the presence of … friction, unrestrained expansion, mixing of two fluids, heat transfer across a finite
temperature difference, electrical resistance, inelastic deformation of solids, and chemical Principal Mode: Abstract-ideal
theorisation
Secondary Modality:
133 Figure 12.4 , p. 474
Figure 12.5, p. 475
reactions… renders a process irreversible” (Cengel & Boles, 2011, p. 291). The physics author, Schroeder (2000), on the other hand, argues that although the irreversibility of processes of the kind described above are not inevitable, they are exceedingly likely, and statistical methods are used to back this up.
Statistical mechanics uses probability theory to explain how microscopic behaviour of large numbers of particles determines the macroscopic properties and behaviour of matter. “… [W]e need to study how systems store energy, and learn to count all the ways that the energy might be arranged” (Schroeder, 2000, p. 49). The mathematical method that allows counting of ways of organising events is combinatorics, and students are introduced to this.
A quantum mechanical model of atoms in a solid is used as the starting point – the Einstein model of a solid (Chabay & Sherwood, 2011; Schroeder, 2000). The authors comment on the usefulness of the model:
[it] allows us to understand in detail the statistical nature of energy transfer between a hot object and a cold object, and why two objects come to ‘thermal equilibrium’… [and] gain a more sophisticated and powerful understanding of the meaning of temperature. (Chabay & Sherwood, 2011, p. 474, emphasis added)
Atoms are modelled as harmonic oscillators:
a solid…[is modelled] as a large number of tiny masses (the atoms) connected to their neighbors by springs (the
interatomic bonds)… We would now like to use this model to ask detailed quantitative questions about the distribution of energy in a solid. (Chabay & Sherwood, 2011, p. 474) The model is simplified even further when each atomic oscillator is pictured as moving independently of the atoms around it in three dimensions, as if it is connected to rigid walls instead of other
oscillating atoms (see Fig 12.4, p.474).
Furthermore, since atoms are considered as independent, a single three-dimensional oscillator can now be replaced by three
independent one-dimensional oscillators (see Fig. 12.5, p.475). This stripping of the model to the barest of detail allows the authors of the textbook to use simpler mathematical modelling of processes, and is an example of Weisberg’s minimalist idealisation, described in chapter
134 two: all extraneous properties of the phenomenon are removed, and only those crucial to the occurrence of the phenomenon are retained. This stripping results in a highly abstract and exceedingly generalised model. The motivation for this type of idealisation (distortion of reality) according to Weisberg (2007a) is the uncovering of the explanatory power of causal factors, as can be seen in what follows here.
The focus is on the sharing of energy, and Chabay and Sherwood start with a small Einstein solid consisting of three quantum oscillators (the equivalent of a single atom), sharing four quanta (units) of energy between them. There are four different general ways (called macrostates) to share 4 quanta of energy between 3 oscillators: either have 4 quanta in one oscillator with none in the other three, or 3 quanta in one, with one quantum in a second and no quanta in a third, or 2 quanta in each of two oscillators with none in the other one, or 2 quanta in one oscillator, and 1 quantum in each of the other two. For each macrostate, there are a number of different ways in which energy can be arranged, and these need to be counted. Each of these represents a microstate (to specify the microstate of a system, the state of every particle in the system has to be specified). Schroeder calls the number of possible microstates for each macrostate the multiplicity Ω of the macrostate:
Macrostate 1: all the energy could be given to one of the three oscillators and none to the other two; there are three ways (three microstates) in which this can be done, multiplicity Ω = 3:
Oscillator 1 Oscillator 2 Oscillator 3
4
0
0
0
4
0
0
0
4
Macrostate 2: Alternatively, 3 quanta can be given to one of the oscillators, 1 quantum to
another and no energy to the third; there are 6 ways (microstates) this can be done, multiplicity Ω = 6:
Oscillator 1 Oscillator 2 Oscillator 3
3
1
0
3
0
1
0
3
1
1
3
0
1
0
3
0
1
3
135 Macrostate 3: Another possibility is for 2 quanta to be given to one of the oscillators, and 1 to each of the other two oscillators; there are 3 ways (microstates) this can be done, multiplicity Ω = 3:
Oscillator 1 Oscillator 2 Oscillator 3
2
1
1
1
2
1
1
1
2
Macrostate 4: The last possible option is for 2 quanta to be given to two of the oscillators, and none to the third; there are three possible ways to do this, multiplicity Ω = 3:
Oscillator 1 Oscillator 2 Oscillator 3
2
2
0
2
0
2
0
2
2
There is therefore a total of 15 possible ways (15 microstates) that 4 quanta of energy can be shared between 3 quantum oscillators. The total multiplicity of the system, Ω = 15.
The fundamental assumption of statistical mechanics is that “over time, an isolated system in a given macrostate is equally likely to be found in any of its possible microstates” (Chabay & Sherwood, 2011, p. 476). Schroeder proves that for N oscillators and q energy units the multiplicity =(𝑞+𝑁−1)!𝑞!(𝑁−1)! .
This is still a long way away from explaining why energy flows from a body at a higher
temperature to one at a lower temperature. To explain how heat flows between two interacting systems, Schroeder describes how energy is shared between two Einstein solids, A and B, with three oscillators each, exchanging 6 units of energy between them. By applying the equation derived above, he shows that there are 462 microstates across 7 macrostates possible. Over a long time scale, all 462 microstates are equally likely since energy is passed randomly according to the fundamental assumption of statistical mechanics (see Figure 2.4, Schroeder, 2000, p. 57).
136 Figure 2.4
Macrostates and multiplicities of a system of two Einstein solids, each containing three oscillators, sharing a total of six units of energy
Figure 2.6
Typical multiplicity graphs for two interacting Einstein solids, containing a few hundred oscillators and energy units (left) and a few thousand (right). As the size of the system increases, the peak becomes very narrow relative to the full horizontal scale. For N ≈ q ≈ 1020, the peak is much too sharp to draw.
However, the summary shows that some macrostates are more probable than others. There are 100 ways in which the energy can be evenly shared between solid A and B, but only 28 ways in which all the energy can be found in solid B. The probability for even sharing of energy is therefore 100/462, whereas for the extreme uneven distribution (all the energy in B) it is only 28/462. This means that if the two solids A and B are in contact with all the energy initially in B and none in A, after some time it is much more likely to find an even distribution of the energy across A and B, and energy has flowed from B to A. When this is scaled up to hundreds of quantum oscillators, Schroeder points out that calculations need to be done by computer, and that the difference between finding the most likely and least likely distribution becomes enormous. For a system with two Einstein solids A and B consisting of 200 and 300 oscillators and 100 quanta energy to distribute, the most likely macrostate is around 1033 times more likely
than the least likely macrostate. If solids A and B were brought into contact for some time, solid B starting out with all the energy initially and A with none, the likelihood that energy has flowed from B to A after some time is now simply astronomical. This implies that the system is
effectively exhibiting irreversible behaviour – energy flows spontaneously from B to A, and never from A to B. This is illustrated in the peak width in the multiplicity graphs below: for
137 systems consisting of large numbers of quantum oscillators and energy quanta, “… out of all the macrostates, only a tiny fraction are reasonably probable” (Schroeder, 2000, p. 60). Schroeder uses Stirling’s approximation to simplify the evaluation of the large factorials, and Gaussian functions to give an indication of the ‘narrowness’ of the multiplicity peak. (See Fig 2.6 Schroeder, p. 60).
He demonstrates that “when two large Einstein solids are in thermal equilibrium with each other, any random fluctuations away from the most likely macrostate will be utterly
unmeasurable” (Schroeder, 2000, p. 66, emphasis in the original). With this the authors of the physics textbooks have now given a microscopic statistical argument to explain the so-called ‘arrow of time’, the fact that macroscopic thermodynamic processes are irreversible. “These considerations show that in the world of macroscopic objects such as ordinary-sized blocks, the most probably arrangement is essentially the only arrangement that is ever observed” (Chabay & Sherwood, 2011, p. 482). Schroeder explains that “[i]rreversible processes are not inevitable, they are just overwhelmingly probable” (2000, p. 49, emphasis in the original).
The description of the use of statistical mechanical methods demonstrates the use of minimalist idealisation in physics to develop abstract-ideal theoretical explanations for the irreversibility of macroscopic processes like the flow of heat. These explanations are powerful in their
universality to apply across contexts. There is nothing specific or particular about the modelling of matter as one dimensional harmonic oscillators. The principal knowledge modality exhibited by the physics knowledge here is idealisation, mode abstract-ideal theorisation, with the
secondary modality the specialisation of the knowledge towards the mode of universals.