A room’s air contains an extraordinary number of moving molecules, yet a thermometer gives a stable reading. A magnet can organise collectively, while a tiny system displays conspicuous fluctuations. Statistical mechanics asks how detailed microscopic possibilities become the larger-scale behaviour we can predict and measure.
The gold standard of statistical mechanics is a probability model whose states, constraints and observables are explicit. It connects microscopic reasoning with macroscopic evidence without pretending that an average is every individual event. This guide develops that connection through original counting, population and random-walk examples, then brings together the existing specialist mechanisms.
The educational purpose follows The Gold Standard of Education: learning that transfers into life. You should be able to explain why a method works, use it in a manageable case and recognise which assumption changes in a different situation. All numerical cases here are constructed teaching models, not experimental measurements or forecasts of an actual material.
Find your way through statistical mechanics
Connect microscopic possibilities with averages, fluctuations, thermodynamics and the limits of equilibrium models.
Full chapter index · Practice and explained answers · How X Works
Chapter contents
Define states and probabilities · Chapters 1–3
Choose the equilibrium model · Chapters 4–7
Connect averages with matter · Chapters 8–11
Explore interactions and learning · Chapters 12–14
A microstate specifies a detailed configuration in a chosen physical model. In a classical particle description it can involve positions and momenta; in a quantum description it involves the appropriate quantum state. A macrostate describes selected larger-scale quantities such as energy, volume or magnetisation. Many microstates can correspond to one macrostate.
The distinction is not simply small versus large objects. It concerns the level of description. Four labelled two-state units already allow a meaningful microstate–macrostate example. A particular assignment, up–down–up–down, is one detailed configuration. The statement two units are up specifies a coarser property shared by several configurations.
The model must say which distinctions count. Labelling units in a teaching example makes its counting transparent. Indistinguishable quantum particles require different state counting. Transferring the labelled counting rule unchanged into that problem can produce an incorrect result even when the arithmetic is flawless.
An observable is a quantity calculated or measured from a state, possibly after an averaging process. The observable might be total energy, particle number or a correlation between locations. Choosing it makes the question concrete: what response is the model supposed to predict?
Before invoking entropy or a distribution, write a sentence naming the system and the information retained. A probability calculation is only meaningful relative to that state space. An unspecified collection of possibilities cannot support a precise physical conclusion just because it has many terms.
CHAPTER 2 OF 18 · Define states and probabilities
2. Count configurations before attaching a slogan
Take four labelled, independent units, each equally likely to be up or down. There are 2⁴ = 16 detailed configurations. The numbers with zero, one, two, three or four up units are 1, 4, 6, 4 and 1. The middle macrostate is more likely because more detailed configurations realise it.
The probability of exactly two up units is 6/16 = 0.375. The probability that all four are up is 1/16 = 0.0625. No individual configuration has been favoured in this model. Unequal macrostate probabilities emerge when we group equally weighted configurations.
Now change the condition: only states with exactly two up units are allowed. The accessible state space contains six configurations, not sixteen. A probability calculated by dividing by sixteen would answer the earlier unconstrained problem. The denominator is a physical modelling choice as well as an arithmetic one.
Change another condition: an external influence makes up more likely than down. The multiplicities remain 1, 4, 6, 4 and 1, but they no longer determine probabilities by themselves. Each configuration now needs its appropriate weight. Counting and weighting are related steps, not synonyms.
This example exposes the central reasoning move. A common outcome need not have an invisible preference built into each microscopic event. It may have many more ways of occurring. But that explanation works only after the state space, constraints and microscopic probabilities have been justified.
CHAPTER 3 OF 18 · Define states and probabilities
3. Entropy describes multiplicity or a distribution
For equally weighted accessible states in a suitable microcanonical description, Boltzmann’s relation is S = kᵦ ln Ω, where Ω counts accessible microstates. For a probability distribution over states, the Gibbs form is S = −kᵦ Σpᵢ ln pᵢ. The natural logarithm and Boltzmann constant give thermodynamic entropy its physical units.
In the four-unit example, a macrostate with six equally weighted configurations has entropy kᵦ ln 6 within the stated description. A macrostate with one configuration has entropy zero relative to that counting model. This does not assert that the real physical object has no other degrees of freedom or a universal absolute entropy of zero.
The word disorder can be an opening analogy, but it is too imprecise to finish the explanation. A visually untidy pattern may have no special thermodynamic significance. Entropy requires a physical state description and the relevant probabilities or multiplicity.
A useful check is to ask which information was discarded when you described the macrostate. If different detailed arrangements are treated as the same macroscopic condition, multiplicity becomes relevant. Changing the description can change the counting question. It does not license arbitrary changes in a thermodynamic prediction while keeping the physical experiment fixed.
The distinction also protects cross-disciplinary transfer. Mathematical expressions resembling entropy occur in information theory, but applying thermodynamic units and conclusions to an unrelated dataset requires more than recognising the same logarithm. Explain the correspondence and its limits rather than treating a shared formula as a complete physical identity.
CHAPTER 4 OF 18 · Choose the equilibrium model
4. Temperature is an equilibrium relationship
Temperature is not simply the total amount of energy in an object. In thermodynamics, inverse temperature is related to how entropy changes with energy under the relevant fixed conditions. Bringing two systems into thermal contact allows energy exchange; equilibrium concerns the combined account.
A large cool object can contain more internal energy than a small hot object. The direction of heat transfer in a simple thermal-contact situation is determined by temperature difference, not by ranking total energy. Keeping those quantities separate prevents a common intuitive error.
A counting model shows the logic. Suppose transferring a small energy packet from system A to B increases the product of their accessible multiplicities. That redistribution is associated with a more numerous combined macrostate. At equilibrium, no further small exchange increases the total entropy in the same way under the stated constraints.
The model does not tell us every exchange’s time history. Equilibrium reasoning and kinetics are different jobs. Two objects may have a thermodynamic drive toward equilibration but exchange energy slowly because their contact is poor. The mechanism and rate of transfer require additional information.
For a learner, connect temperature first with comparison and equilibrium, then with microscopic models appropriate to particular systems. In a dilute classical gas it relates to average translational kinetic energy, but that familiar result is not a definition that can be copied unchanged into every interacting or quantum system. State the model that makes the link valid.
CHAPTER 5 OF 18 · Choose the equilibrium model
5. An ensemble specifies the allowed exchange
An ensemble is a probability description reflecting constraints and environmental contact. A microcanonical model fixes energy, volume and particle number. A canonical model fixes temperature, volume and particle number while allowing energy exchange with a reservoir. A grand-canonical model allows particle exchange as well, with chemical potential playing a role.
These are descriptions of physical questions, not three competing brands of statistics. Choose according to what is controlled and what can fluctuate. A small subsystem in thermal contact with a much larger environment can have fluctuating energy even while the larger combined system has a fixed total energy.
Imagine a sealed collection of two-state units exchanging energy with a reservoir but not particles. A canonical description can be appropriate. Opening a path that permits particles to enter and leave changes the modelling job. Keeping particle number fixed in the calculation may then erase the very fluctuations under investigation.
In many large ordinary equilibrium systems, ensemble predictions agree for suitable bulk observables in an appropriate limit. That useful agreement is not permission to ignore constraints in every finite system, long-range-interaction problem or unusual regime.
A good first line in a solution is therefore a boundary statement: what is the system, what can cross its boundary and which quantities are held fixed? That sentence often determines the correct distribution before any sophisticated mathematics is needed. It also makes the answer reviewable by someone who was not present when the model was chosen.
CHAPTER 6 OF 18 · Choose the equilibrium model
6. Boltzmann weights become probabilities after normalisation
For a canonical equilibrium model, a state of energy Eᵢ has probability proportional to exp(−Eᵢ/kᵦT). Lower-energy states receive greater weight, but the number of states at each energy also matters. A highly populated energy level can owe its importance to degeneracy rather than to low energy alone.
The partition function Z = Σexp(−Eᵢ/kᵦT) sums state weights and normalises the distribution. Then pᵢ = exp(−Eᵢ/kᵦT)/Z. If you sum over energy levels rather than individual states, include the number of states associated with each level.
David Tong’s statistical-physics fundamentals develops the formal canonical framework, partition function and thermodynamic connections. Here the immediate learning task is to distinguish a raw weight from a probability. A weight need not sum to one; the normalised probabilities must.
An arbitrary common shift in all energies changes each weight by the same factor. The partition function changes by that factor too, leaving the probabilities unchanged. This is a helpful consistency check: moving the zero of energy should not change the relative occupations of unchanged states.
The equilibrium assumption remains essential. A driven, rapidly changing or insufficiently relaxed system need not follow this distribution. A measured population that differs from a Boltzmann prediction may expose missing degeneracy, incorrect temperature, interactions or lack of equilibrium. The discrepancy is a question to investigate rather than an automatic refutation of probability.
CHAPTER 7 OF 18 · Choose the equilibrium model
7. A two-level calculation makes the model tangible
Consider independent units with one ground state at energy zero and one excited state at energy ε. Choose a teaching temperature at which ε/kᵦT = ln 3. The ground-state weight is 1 and the excited-state weight is 1/3. The one-unit partition function is 4/3, so the excited-state probability is 1/4.
For 120 independent units, the expected number excited is 30. Expected total excitation energy is 30ε. That expectation does not mean every observation contains precisely 30 excited units. The population fluctuates, and the independence assumption defines its distribution.
Now change the context: there are three distinct excited states at the same energy ε, with one ground state. The total excited-level weight becomes 3 × 1/3 = 1. Its probability is 1/(1 + 1) = 1/2. The energy gap and temperature stayed the same, but degeneracy changed the prediction.
This original example is a useful diagnostic because it reveals whether the learner is counting states or merely inserting an energy gap into a remembered expression. A statement about an energy level is incomplete when several states share that energy.
At very high positive temperature relative to the gap, weights per state approach equality in this finite model. In the nondegenerate case the excited fraction approaches one half; in the threefold-degenerate case it approaches three quarters. The asymptotic check provides another way to catch a missing degeneracy factor.
CHAPTER 8 OF 18 · Connect averages with matter
8. Averages and fluctuations answer different questions
For 120 independent two-state units with excited probability p = 1/4, the excited-count mean is Np = 30 and variance is Np(1 − p) = 22.5. The standard deviation is approximately 4.74. A single count of 34 is therefore not, by itself, an alarming disagreement with the mean.
Scale the model to 12,000 otherwise identical independent units. The mean becomes 3,000 and standard deviation approximately 47.4. Absolute fluctuations are larger, but relative fluctuations are smaller: about 1.58% instead of 15.8%. The stable macroscopic average can coexist with microscopic randomness.
That square-root behaviour relies on the model. Correlated units, proximity to a critical point or different probability structure can change the fluctuations. A large number alone does not establish independence. It is particularly important not to present 1/√N as a universal uncertainty law for every physical system.
A practical data check compares both location and spread. A model can predict the average adequately while missing correlations or variance. Two datasets with the same mean can reflect very different mechanisms. The fluctuation–dissipation guide in the directory shows how fluctuations can themselves carry information about response under appropriate conditions.
The learning transfer is precise: whenever a statement says the average is stable, ask what distribution and correlations make it stable. Do not assume that one central value describes every event, every short time interval or every region of the sample.
Correlation can preserve the mean and change the spread
Compare two original models, each containing eight two-state units with equal up and down probabilities. In model A all eight units are independent. The number up has mean four and variance eight times one half times one half, which is two. Many different counts are possible.
In model B the units form four independent pairs, but the two members within each pair always match. Each pair is equally likely to be up–up or down–down. The total up count still has mean four. Write it as twice the number of up pairs: its variance is four times the variance of that pair count, or four. The same mean now comes with twice the variance.
| Model | Mean number up | Variance of number up | Possible count pattern |
|---|---|---|---|
| Eight independent units | 4 | 2 | Every integer from 0 to 8 |
| Four perfectly matched independent pairs | 4 | 4 | Only even counts from 0 to 8 |
A measurement reporting only the average would not distinguish these models. Repeated observations of the distribution could. Seeing only even counts would be a particularly direct clue in this deliberately exact example, although real measurements may add noise or imperfect pairing.
The calculation also shows why treating every observed unit as an independent contribution can underestimate uncertainty. The effective information is shaped by dependence. This matters when interpreting correlated regions of matter, time samples with memory or simulation data produced by slowly evolving configurations.
The analogy should not be stretched into a universal formula for every correlated system. Real correlations have strengths, distances and timescales, and their contribution must be modelled or measured. What transfers is the question: which units can vary independently, and which tend to move together? Asking that before applying a square-root rule makes the statistical account much more trustworthy.
CHAPTER 9 OF 18 · Connect averages with matter
9. Gas pressure connects collisions with a bulk quantity
In a dilute classical-gas picture, particles transfer momentum when they encounter a boundary. The aggregate effect produces pressure. The ideal-gas relationship links pressure, volume, particle number and temperature within assumptions that neglect important interaction and finite-size effects in the appropriate regime.
The Feynman lecture on kinetic theory develops this collision-based connection. Its value for a beginner is that a macroscopic force becomes intelligible through microscopic motion, rather than appearing as a separate unexplained rule.
A simple proportionality test is useful. At fixed temperature and particle number, halving the volume doubles ideal-gas pressure. At fixed volume and particle number, doubling absolute temperature doubles pressure. Doubling a Celsius reading does not generally double absolute temperature: convert to kelvin before applying the proportionality.
Now change the physical conditions until interactions, condensation or very high density matter. The ideal equation may no longer describe the material adequately. The successful simple model is a starting reference against which departures can be investigated, not a promise that every gas at every pressure behaves ideally.
The important step is to connect each controlled quantity with a prediction. If the container leaks while you compress it, particle number is no longer fixed. The original proportionality now describes a different experiment. A boundary account protects both the mathematics and the interpretation.
CHAPTER 10 OF 18 · Connect averages with matter
10. Random walks explain spreading without a preferred direction
Take an ideal one-dimensional random walk with independent steps of length a, equally likely left or right. After n steps, the expected displacement is zero, but the mean squared displacement is na². The root-mean-square displacement is a√n. Zero mean displacement does not mean no spreading.
In an original example with a = 2 micrometres and n = 100, the root-mean-square displacement is 20 micrometres. After 400 steps it is 40 micrometres. Four times as many steps doubles this characteristic spread rather than quadrupling it.
If each step takes the same interval and the assumptions connect the walk with diffusion, the one-dimensional relation is mean squared displacement = 2Dt. In three independent spatial dimensions, the corresponding radial mean square is 6Dt. State which dimensional quantity is being calculated before comparing formulae.
Now introduce a preferred direction. The average displacement can become nonzero while random spreading continues around it. Drift and diffusion are different contributions. A moving centre does not eliminate variability, and variability does not establish that the average must remain fixed.
This example connects naturally with molecules, suspensions and some biological transport models. The transfer has limits: active motion, confinement, memory and heterogeneous environments can break the elementary independent-step assumptions. An observed departure from square-root spreading should prompt a model review rather than an automatic attempt to force the same diffusion coefficient through every timescale.
CHAPTER 11 OF 18 · Connect averages with matter
11. Free energy selects the appropriate equilibrium account
At fixed temperature, volume and particle number, Helmholtz free energy F = U − TS is the relevant equilibrium potential. At fixed temperature and pressure in an appropriate closed-system setting, Gibbs free energy serves the corresponding job. The choice reflects the constraints, not a preference for one impressive formula.
Free energy brings energy and entropy into the same account. A state with lower energy can compete with another state realised by more microscopic configurations. Temperature changes the relative significance of the entropy contribution. The phrase systems seek lower energy alone is therefore incomplete for many thermal-contact situations.
Consider two hypothetical macrostates with energy difference ΔU = 2 units and entropy difference TΔS = 3 units at a specified temperature. Their Helmholtz difference is ΔF = 2 − 3 = −1 unit for the second relative to the first. Under the stated constraints, the entropic contribution outweighs the energy cost.
Change temperature and the comparison may change, provided the state properties and their temperature dependence are modelled appropriately. Holding every term constant while changing T can be a teaching simplification; it should not be mistaken for a full phase diagram of a real substance.
The useful reasoning habit is to identify the environment before judging a preferred state. A result derived at fixed volume cannot be transferred silently to a fixed-pressure problem with volume work ignored. Equilibrium is a conditional comparison of complete systems and constraints.
CHAPTER 12 OF 18 · Explore interactions and learning
12. Interactions produce phases and correlations
Independent units are a helpful starting point because their probabilities are manageable. Interacting units can organise collectively. The probability of one configuration can depend on its neighbours, and fluctuations can become correlated over larger distances.
The Ising model provides a well-known simplified language for such questions: local variables interact according to stated rules. Its purpose is not to reproduce every detail of a real magnet. It can isolate relationships between interactions, dimensionality, temperature and collective order.
An original three-unit example makes dependence visible without advanced mathematics. If neighbouring units are energetically rewarded for matching, the all-up and all-down configurations can carry different weights from mixed configurations even at zero external field. The mean magnetisation of a symmetric finite ensemble may still be zero while correlations favour alignment.
That distinction repairs a subtle misconception. A zero mean does not prove the absence of organised tendencies. Examine correlations, distributions and the conditions of observation. A system can switch between states whose average hides their individual structure.
The critical-phenomena and percolation guides below extend the question in different directions. Universality concerns shared large-scale behaviour under particular conditions; it does not mean every system has the same microscopic mechanism. The challenge is to explain which details matter at the chosen scale and which can be omitted while preserving the observable being studied.
CHAPTER 13 OF 18 · Explore interactions and learning
13. Equilibrium does not describe every time-dependent process
Statistical mechanics also studies systems driven away from equilibrium, currents, relaxation and entropy production. A stationary-looking average does not guarantee equilibrium. A system can maintain a constant profile because an external drive continually supplies and removes energy or particles.
Imagine an invented channel with a constant average concentration profile while particles flow through it. The profile’s apparent stability does not mean exchange has stopped. It is a steady-state transport problem whose boundary conditions support continuing current.
Likewise, a cooling experiment may cross a transition faster than the system can relax. The observed state can retain memory of its history. The Kibble–Zurek and supercooling guides explore particular aspects of dynamics and delayed reorganisation; they should not be treated as interchangeable labels for any slow response.
The fluctuation-theorem guide concerns carefully defined statistical relations in specified settings. It does not license the claim that ordinary macroscopic processes routinely reverse without cost, nor that an isolated brief fluctuation abolishes the second law. Probabilities, observation intervals and the complete system account remain essential.
For transfer, label a diagram with what enters, what leaves and what changes over time. Then decide whether equilibrium, steady state or transient behaviour is the intended model. This simple classification can prevent a sophisticated equilibrium calculation from being applied to a process maintained by an unacknowledged external drive.
CHAPTER 14 OF 18 · Explore interactions and learning
14. Learn the subject by making assumptions inspectable
The prerequisite difficulty is often not calculus alone. Students need fluency with probabilities, logarithms, units and the meaning of a constraint. A learner may differentiate a partition function correctly yet sum over the wrong states. The repair begins with the state space, not another derivative exercise.
Teach counting through small models that can be listed explicitly. Then add weights and normalisation. Compare distributions with the same mean but different spread. Only after those distinctions are secure should compact formal expressions compress the reasoning.
A useful worked-to-independent sequence starts with the four-unit model. Let the student count configurations, then constrain the number of up units, then change their probabilities. Ask which part of the calculation changes each time. The learner can see the denominator and weights as consequences of the model rather than arbitrary formula choices.
Computation helps when it exposes the same structure. Simulate repeated independent trials and compare their distribution with the expected mean and variance. Change the random seed or run length. A single smooth histogram is not a proof, and a noisy small sample is not a contradiction of the model.
Professional work adds deeper mathematics, experiment, simulation, numerical convergence and knowledge of the specific physical system. Its transferable habit is to make assumptions visible enough that someone else can test them. A compact elegant formula becomes useful when its state space, constraints and observable can be reconstructed.
CHAPTER 15 OF 18 · Check and explore
15. Common questions and misconception repairs
Is statistical mechanics just ordinary statistics applied to physics?
It uses probability and statistics, but its central task connects physical states, interactions and constraints with thermodynamic and dynamical behaviour. The physical model determines the probabilities and observables being investigated.
Does entropy mean everything must look messier?
No. Visual disorder is an incomplete analogy. Entropy concerns multiplicity or a probability distribution within a physical description. Local organisation can develop while the complete system and environment follow the relevant thermodynamic account.
Does the most likely macrostate occur every time?
No. Probability is not a schedule. The distribution, system size, correlations and observation conditions determine how substantial fluctuations can be. A single outcome should be compared with a distribution rather than treated as identical to the mean.
Can a finite magnet have zero average while favouring alignment?
Yes, in an appropriate symmetric model. Opposite organised configurations can cancel in the average. Correlations or the full distribution can reveal structure that the mean alone hides.
Is a partition function a physical box of particles?
No. It is a sum of statistical weights for a defined model and ensemble. Its usefulness comes from its connections to probabilities and thermodynamic quantities, not from being a new microscopic object.
Why are different ensembles needed?
They represent different constraints and exchanges. Choosing the ensemble is part of specifying the physical problem. Large-system equivalences in suitable cases do not remove the need to understand the boundary.
CHAPTER 16 OF 18 · Check and explore
16. Practice with answers and changed contexts
First, six labelled independent units each have two equally likely states. There are 64 configurations. Exactly three up units can occur in 20 ways, giving probability 20/64 = 0.3125. If exactly three up is imposed as a constraint, the accessible set instead contains 20 states and that macrostate is certain within the constrained model.
Second, one ground state has weight 1 and two excited states each have weight 1/4. The partition sum is 1.5 and total excited probability is 0.5/1.5 = 1/3. Giving one excited state the probability 1/4 without normalisation would confuse a weight with a probability.
Third, 200 independent units have excited probability 0.10. The expected excited count is 20 and variance is 18, giving standard deviation about 4.24. Doubling the population doubles the mean and variance but multiplies standard deviation by √2. State independence before using that scaling.
Fourth, an ideal unbiased walk has root-mean-square displacement 3 mm after one minute. Under the same diffusion model, reaching 6 mm requires four minutes, not two. If the model includes drift, the mean displacement and spreading must be treated separately.
Finally, a system maintains a constant temperature profile while a heater supplies energy and a cooler removes it. That can be a nonequilibrium steady state. A constant profile alone does not establish equilibrium. Identify the continuing energy current and boundary conditions before choosing an equilibrium ensemble.
CHAPTER 17 OF 18 · Check and explore
17. Connected mechanisms grouped by learning job
Start with the physical and probabilistic foundations
Statistical mechanics mechanism guide develops microstates, ensembles and thermodynamic emergence. The Gold Standard of Thermodynamics connects heat, work and macroscopic constraints. The Gold Standard of Statistics develops uncertainty and inference. The Gold Standard of Quantum Mechanics provides the state language needed when classical counting no longer applies.
Follow fluctuations, spreading and response
Fluctuation–dissipation connects spontaneous variation with response under stated conditions. Fluctuation theorems examines carefully defined small-system statistics. Thermal transpiration follows a transport effect involving temperature differences and confinement. Biophysics connects diffusion and thermal motion with living systems. These are distinct applications of physical probability, not one interchangeable mechanism.
Explore collective changes and dynamics
Percolation examines the appearance of connected pathways. Critical phenomena explores shared large-scale behaviour near transitions. Superfluidity develops collective quantum flow. The Kibble–Zurek mechanism connects transition dynamics with defect formation. Supercooling examines delayed transformation and the role of nucleation. The glass transition introduces slow relaxation and preparation history. The related condensed-matter mechanisms explain why interactions and timescales can make independent-unit models inadequate.
Extend models into neighbouring subjects
Condensed matter physics follows collective material properties. Physical chemistry connects molecular motion, energy and kinetics. The Gold Standard of Computational Chemistry examines model predictions and numerical evidence. The Gold Standard of Biophysics explores physical models of living systems. The Gold Standard of Education and How X Works connect these subjects through explanation and transfer.
CHAPTER 18 OF 18 · Check and explore
18. Sources, boundaries and a final explanation
Primary teaching references checked on 21 October 2026 include David Tong’s statistical-physics course and its fundamentals chapter, alongside the Feynman kinetic-theory lecture. These provide formal routes into equilibrium distributions and the microscopic account of gas pressure. The specialist ecosystem articles provide further sources for their individual mechanisms.
The counting tables, selected energy ratios, population sizes and random-walk distances in this article are original pedagogical constructions. Their purpose is to expose state counting, normalisation, independence and boundary choices. They are not measurements of a real magnet, gas or biological system, and their approximations are not universal descriptions of matter.
For one final task, explain how an average can remain stable while its microscopic contributors fluctuate. Name the units, their probabilities, the observable and the independence or correlation assumptions. Then change one assumption and say which part of the prediction needs revision.
That explanation carries the subject into life without turning every human or social system into a gas. Statistical mechanics teaches a disciplined way to connect levels of description. Its gold standard is a macroscopic claim whose microscopic model and probability account remain visible enough to question, calculate and test.