A gas can contain mobile electrons and ions that respond together to electric and magnetic fields. A local disturbance then changes the fields, and those fields influence many other particles. This feedback produces behaviour that cannot be understood by following isolated collisions alone.
Plasma physics studies these charged, collective systems: their motion, waves, transport, instabilities and interactions with boundaries. Its gold standard identifies the relevant scales, chooses a model that fits them and checks the prediction against measurements. A bright glow or a high temperature is an observation, not a complete explanation of the plasma.
The original numerical examples below are ideal teaching cases, not specifications for an experiment or forecasts of a fusion device. They follow The Gold Standard of Education: learning that transfers into life: understand the mechanism, use it and revisit the assumptions when the situation changes.
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Begin with the route that fits your purpose. Read sequentially for the full argument, or use the chapter index to return to a worked example, an evidence check or a connected guide.
Full chapter index · Plasma Physics: mechanism library · How X Works
Chapter index
Understand the question · Chapters 1–3
Build the mechanisms · Chapters 4–7
Work through evidence · Chapters 8–11
Use and check the ideas · Chapters 12–15
Transfer and continue · Chapters 16–19
In an ionised gas, some electrons are separated from neutral atoms or molecules. The resulting mixture may contain electrons, positive ions, neutral particles and, in some settings, additional charged components. The degree of ionisation varies widely between environments.
Mobile charges respond to fields and contribute to them. If electrons move away from ions in a small region, the resulting charge separation creates an electric field that influences subsequent motion. This self-consistent response is central to plasma behaviour.
The phrase “fourth state of matter” is a useful introduction, but it should not imply that every plasma is the same kind of extremely hot gas. Electrons and heavier particles can have different characteristic energies. Some technological plasmas are only partly ionised, with relatively cool bulk gas.
The US Department of Energy’s plasma overview introduces this variety and its scientific and technological relevance. The common thread is charged-particle behaviour and collective fields, rather than one universal temperature or appearance.
Start an investigation by listing the components: which ions, how many neutrals, approximate electron density, temperatures and imposed fields. Then identify the question: wave propagation, particle confinement, surface processing or energy transfer. These details determine which physical effects must be included. A single label such as “plasma” cannot choose the model on its own.
A model needs a spatial and temporal range. A particle gyration, a collective oscillation and a slow evolution of an entire device occur on different scales. Treating them as interchangeable can produce equations that look sophisticated while answering the wrong question.
Useful lengths include the system size, shielding length, particle gyroradius and mean free path. Useful times include oscillation periods, collision times, transit times and the duration over which a disturbance grows. Ratios between these scales help determine which approximations are sensible.
For example, a fluid description averages over detailed particle motion. It can be powerful when the quantities and closures used adequately describe the process. A kinetic description retains information about particle velocity distributions that may matter when collisions are weak or resonant interactions are important.
Neither description is automatically more truthful for every task. A more detailed calculation with poorly known inputs can be less useful than a simpler model with well-tested assumptions. The job is to include the detail that controls the observable being investigated.
A beginner can practise this without solving advanced equations. Given a system size and a microscopic length, calculate their ratio and explain the implication. If the lengths are comparable, a large-scale approximation may fail near the feature of interest. If they are widely separated, averaging may be reasonable, subject to the other assumptions. This habit turns model selection into a physical argument rather than a preference for the most impressive formula.
Many plasmas have nearly balanced positive and negative charge densities over lengths large compared with their shielding scale. This approximate balance is called quasineutrality. For a simple singly ionised plasma, electron and ion number densities are then approximately equal in the bulk.
Approximate is important. A small imbalance can produce an electric field, and boundaries or rapid disturbances can require a more explicit charge-separation description. Quasineutrality should not be imposed at every point and every scale merely because the system is a plasma.
Nor does balance eliminate currents. Electrons and ions can move differently while their average densities remain nearly matched. Current depends on charge motion; charge density depends on the local balance of charges. Confusing the two makes a conducting plasma seem paradoxical when it is not.
Think of two equally populated streams moving in different ways. Equal population does not imply equal transport. The analogy helps separate number balance from motion, although actual plasma currents require the signs and velocities of each species to be included.
For a practical check, ask three separate questions: is net charge density small in this region; is current present; and is the electric field negligible for the process? The answers need not be identical. A model that assumes all three from one observation can miss important dynamics. Learning to keep these quantities separate is an early step towards reasoning about plasma circuits, space environments and confinement systems accurately.
When a charge perturbs a suitable plasma, mobile charges rearrange and reduce its electrostatic influence over distance. In a simple thermal-electron model with a suitable background, the electron Debye length is λD = √(ε₀kBTₑ/(nₑe²)). Here ε₀ is vacuum permittivity, kB is Boltzmann’s constant, Tₑ is electron temperature, nₑ is electron density and e is the elementary charge.
The expression reveals useful dependencies before any number is inserted. At fixed density, quadrupling electron temperature doubles λD. At fixed temperature, quadrupling density halves it. Density strengthens collective shielding; thermal motion resists a concentrated rearrangement.
Consider two ideal plasmas A and B with equal electron temperature. B has nine times A’s electron density. Then λD,B/λD,A = √(1/9) = 1/3. If A’s shielding scale is 0.6 millimetres, B’s is 0.2 millimetres under the model.
These results do not establish that either system satisfies every condition needed for a classical plasma treatment. The system size, number of particles involved, collision behaviour and distribution functions also matter. Near a boundary or in a strongly coupled regime, a simple textbook expression may not describe the full response.
Richard Fitzpatrick’s University of Texas plasma notes on shielding provide a derivation-oriented reference. The teaching example here uses the proportionality to expose the reasoning. The important move is to connect the formula with the physical competition and then state which quantities were held constant.
Displacing electrons relative to ions can create a restoring electric field. In a simple electron oscillation model with ions treated as a stationary background, the angular plasma frequency is ωpe = √(nₑe²/(ε₀mₑ)). The ordinary frequency is fpe = ωpe/(2π).
The distinction between angular frequency and cycles per second matters. A numerical value in radians per second cannot be relabelled hertz without the factor of 2π. Unit discipline prevents a predictable error that can otherwise survive several lines of correct algebra.
At fixed composition in this ideal model, quadrupling electron density doubles ωpe. Temperature does not appear in this simplest frequency expression, although finite-temperature effects matter in more detailed wave descriptions. Compare that with Debye length, which depends on both density and temperature.
For a changed-context exercise, increase density by a factor of four and temperature by a factor of four. The electron Debye length remains unchanged because Tₑ/nₑ is constant, while plasma frequency doubles because density increased. Equal shielding length therefore does not mean equal dynamical response.
The University of Texas discussion of plasma frequency connects the collective timescale with the physical response. The lesson is to use more than one parameter when characterising a system. Two plasmas can agree on one convenient number while differing on the timescale relevant to the measurement or control problem.
For a nonrelativistic charged particle, the Lorentz force is q(E + v × B). The magnetic part is perpendicular to velocity and therefore does no work on the particle directly. It changes direction rather than kinetic energy in the ideal magnetic-force-only case.
In a uniform magnetic field, motion perpendicular to the field gives circular gyration with radius rL = mv⊥/(|q|B). The angular cyclotron frequency is |q|B/m. Motion parallel to the field is unaffected by the magnetic force in this simplest setting, producing a helical path when both components are present.
Take an ideal ion with fixed mass, charge and perpendicular speed. Doubling the magnetic field halves its gyroradius and doubles its cyclotron frequency. Doubling the speed at fixed field doubles the radius but leaves that nonrelativistic frequency unchanged.
An electron and ion with the same perpendicular kinetic energy do not have the same gyroradius. Using v⊥ = √(2K⊥/m), the radius becomes √(2mK⊥)/(|q|B). At equal charge magnitude, field and energy, the heavier particle has the larger radius.
The conditions matter: relativistic effects change the simple relationship, real fields may vary and collisions can interrupt motion. A magnetic field also does not provide unlimited confinement by itself. Field geometry and motion along field lines matter. The worked relationship is a local building block; understanding an entire plasma requires connecting it to collective fields, boundaries and transport.
In crossed electric and magnetic fields, a charged particle’s guiding centre can drift. For uniform fields in the ideal regime, the E × B drift velocity is E × B/B². Its direction is perpendicular to both fields and is the same for positive and negative charges under the shared assumptions.
For perpendicular fields with electric-field magnitude 100 volts per metre and magnetic-field magnitude 0.05 tesla, the drift-speed magnitude is E/B = 2,000 metres per second. The units reduce correctly because a tesla is a volt-second per square metre.
Now double B while holding E fixed. Drift speed halves to 1,000 metres per second. This differs from the cyclotron frequency, which increases with B. “A stronger field makes everything faster” is therefore not a valid general rule. Different motions depend on the field differently.
Field gradients and curvature introduce other drifts, and collisions or time variation can change the simple picture. A real transport calculation may require several effects together. Guiding-centre approximations also need scale separation between gyration and the variation being modelled.
The practical learning move is to draw three arrows: E, B and the predicted drift direction. Then state the assumed geometry and compare with the gyration around the guiding centre. A diagram separates two motions that are easily blended into one imagined spiral. The result becomes transferable when the learner can rotate the field arrangement and recover the new direction rather than remember one orientation from a textbook picture.
A gas description often assumes that collisions help establish a familiar thermal distribution. Plasmas can have circumstances where electrons, ions and neutrals exchange energy at different rates, so one temperature may not adequately describe every component.
Temperature characterises a distribution of random motion under an appropriate definition. It is not the bulk flow speed. A fast-moving population can have a narrow spread around its mean, while a population with little mean motion can have substantial thermal spread.
Scientists often express kBT in electronvolts. This is an energy scale associated with temperature, not a claim that temperature and electric potential are identical quantities. An electronvolt is the energy gained by an elementary charge moving through a potential difference of one volt.
Collision frequency and mean free path help indicate whether local fluid assumptions are appropriate. But rare collisions do not imply absent interactions: collective electric and magnetic fields can still strongly influence motion. The plasma can exchange energy through mechanisms that are not captured by a simple sequence of binary impacts.
For a useful comparison, imagine two populations with the same density and mean energy but different velocity-distribution shapes. A resonant interaction may affect them differently because it depends on the particles near a particular velocity. An average can hide the feature controlling the process. This is why kinetic theory is not merely a more complicated version of the same fluid variables; it can retain information that the chosen averages have discarded.
Plasmas support several wave families because charge separation, pressure and magnetic stresses can all provide restoring effects. The appropriate mode depends on field direction, frequency, wavelength and the properties of the components.
Alfvén waves provide a useful large-scale example. In a suitable magnetohydrodynamic description, magnetic tension acts with plasma inertia. The characteristic Alfvén speed is vA = B/√(μ₀ρ), where ρ is mass density and μ₀ is vacuum permeability.
At fixed field, quadrupling mass density halves vA. At fixed density, doubling field doubles it. These relations describe a modelled propagation scale; they do not identify every observed oscillation as an Alfvén wave.
To identify a wave, compare frequency, wavelength, polarisation, propagation direction and relationships between fluctuating quantities. A repeating signal can also come from an instrument, a rotating structure or an imposed source. The pattern needs a mechanism, not merely a familiar wave label.
Landau damping is a particularly instructive kinetic example: a wave can exchange energy with particles through resonant interaction even without ordinary collisional dissipation. Its interpretation depends on the velocity distribution and the wave. The existing specialist guide develops that mechanism separately.
The transfer check is to ask which restoring force and which inertia a proposed wave uses. Change density or field and predict the direction of the response before calculating. Then ask what observation could distinguish that wave from another mode. This sequence connects equations, physical explanation and diagnostic evidence instead of treating every plasma oscillation as one generic vibration.
A disturbance is unstable when its amplitude grows under the relevant dynamics. The growth may draw on pressure gradients, relative flows, currents or other sources of free energy. Naming an instability should identify the energy source and conditions, not simply describe an experiment that behaved inconveniently.
In a linear teaching model, amplitude A(t) = A₀exp(γt), with positive growth rate γ. If γ = 200 per second, the e-folding time is 1/γ = 0.005 seconds, or 5 milliseconds. An e-folding multiplies amplitude by e, approximately 2.718; it does not double it.
The doubling time is ln(2)/γ, approximately 3.47 milliseconds. Starting from amplitude one in arbitrary units, the linear model gives exp(2) ≈ 7.39 units after 10 milliseconds. Keep amplitude distinct from energy, which often scales with amplitude squared for a defined perturbation.
Linear growth cannot usually be extrapolated indefinitely. As a disturbance grows, it changes the background, interacts with other modes and reaches nonlinear behaviour. Saturation, transport or disruption require additional analysis. A growth rate is not a prediction of the final amplitude by itself.
A useful evidence test compares the measured early-time rate with the proposed mechanism, then asks whether the background conditions match the model. If several instabilities are possible, changing one control parameter may help discriminate between them. The scientific objective is to understand the route from available energy to observed change, not merely to fit an exponential somewhere in a noisy record.
Magnetic fields store energy, and their configuration matters. In magnetic reconnection, the connectivity of magnetic field lines changes in a region where the ideal frozen-in description no longer holds. This can support rapid transfer of magnetic energy into plasma motion, heating and energetic particles.
Field lines are a representation of field structure, not physical strings that are literally cut by a pair of microscopic scissors. The string analogy can illustrate connectivity but should not replace the electromagnetic and plasma dynamics.
Turbulence involves interacting fluctuations over a range of scales. It can alter transport and redistribute energy in ways that a smooth average-field model misses. Observing irregular motion does not by itself determine the energy-transfer pathway or establish a universal diffusion coefficient.
Measurements need to resolve relevant spatial and temporal scales. A single probe can confuse a structure passing the instrument with a change occurring locally. Multiple measurements, model comparisons and assumptions about motion help interpret the record.
These processes connect laboratory and space plasmas, but similar vocabulary does not imply identical conditions. Density, collisionality, magnetic geometry and accessible scales can differ greatly. A laboratory experiment can test a particular mechanism or scaling relationship without reproducing an entire solar event. The useful cross-context question is which dimensionless parameters and physical balances are shared, and which remain different enough to limit the comparison.
Confinement asks how long particles or energy remain in a useful region. Heating, losses, transport and boundaries all contribute. A bright or hot plasma at one instant does not establish that energy is retained for a long time.
A simple energy-confinement scale is τE = W/Ploss, where W is stored plasma energy and Ploss is the relevant loss power under a defined accounting convention. In a steady state, heating balances losses; during a transient, the change in stored energy must also be included.
For an original ideal case, stored energy is 6 megajoules and loss power is 3 megawatts. Since a megawatt is a megajoule per second, τE = 2 seconds. If stored energy remains 6 megajoules but losses rise to 6 megawatts, the scale falls to one second.
This does not mean every particle stays for exactly two seconds. Particle and energy confinement are different quantities, and a real device may have several transport channels and time-dependent terms. State what was measured and which power components were included.
Magnetic confinement also depends on geometry and stability, not just field strength. Charged particles can move along field lines, and collective processes can carry energy across them. Understanding a confinement result therefore requires the connection between local motion and the system’s overall balance. This is where the earlier particle, wave and instability ideas become parts of one physical argument.
Fusion combines light nuclei under suitable conditions, with energy release depending on the reaction. Plasma physics helps investigate how to create and sustain the necessary particle distributions, temperatures and confinement. Nuclear reactions, materials and engineering add further requirements.
The Department of Energy’s fusion-energy-science overview describes this multidisciplinary field. A particular plasma result should be reported at its actual boundary: fusion energy produced, heating delivered to the plasma, facility electricity consumed or electricity exported are different quantities.
Consider a purely fictional comparison. A plasma receives 10 units of external heating and produces 20 units of fusion energy, giving a ratio of two at that boundary. If a facility consumes 80 units of electricity and recovers only 6 units as electrical output, the facility’s electrical balance is negative. Neither number is contradictory; they use different inputs and outputs.
The example is not a description of a real experiment or a prediction of commercial readiness. It demonstrates why a performance ratio must travel with its definition. Pulse duration, repetition, reliability, fuel handling and material behaviour also affect whether a scientific result can support a practical energy system.
The everyday transfer is to read “gain” and “efficiency” carefully. Ask which energy forms cross which boundary, over what interval and with which auxiliary systems included. A headline can be technically accurate at one boundary yet insufficient to answer a household or grid-level question. Better understanding begins by preserving the distinction.
Plasma properties are inferred through instruments with different sensitivities. Spectroscopy can reveal information about emitting species and energy distributions. Interferometry can constrain an integrated electron density. Probes can sample local electrical response under appropriate conditions. No single instrument measures every desired property directly.
A line-of-sight measurement is an integral through a volume, not automatically a local central value. Recovering spatial structure may require several views, an assumed symmetry or an inversion model. Report those assumptions when translating the signal into a profile.
Similarly, a broad spectral line can have several contributions. Thermal motion, instrumental response and other effects need to be separated before attributing all width to one temperature. A good fit with one convenient mechanism is not necessarily a unique physical explanation.
Calibration, timing, uncertainty and instrument perturbation belong in the evidence chain. A probe can affect the plasma it samples, and a measurement of electrons need not establish ion temperature. Compare complementary diagnostics where the conclusion requires more than one component.
Professional practice links experiment, theory, computation and engineering. Preserve configuration, calibration, processing choices and model versions so others can assess the result. Simulations also need convergence checks and comparisons with appropriate observations. A visually convincing plasma animation is not validation. The gold standard is a traceable relationship between a defined property, a measured signal and the model used to connect them.
Begin with charge, force, energy and vectors. Then distinguish single-particle motion from collective response. Introduce shielding and oscillation scales before combining waves, transport and confinement. This gives technical terms a physical job at each stage.
A strong worked example names the assumptions before the equation. After solving, change one condition: density, temperature, magnetic field, angle or boundary. Ask for a qualitative prediction, then calculate. This exposes whether the learner understands the dependency or only recognises a familiar arrangement of symbols.
Classify errors precisely. Confusing angular frequency with hertz requires a units repair. Assuming quasineutral means zero current requires a concept repair. Applying a linear instability model at unlimited amplitude requires a model-validity repair. Treating plasma gain as net electricity requires a boundary repair.
For a final independent task, compare two fictional plasmas. B has four times A’s density, the same electron temperature and twice the magnetic field. Under the earlier ideal models, B has half the electron Debye length, twice the electron plasma frequency and half the gyroradius for the same test particle speed. Several changes can therefore occur together without collapsing into one general “better plasma” ranking.
Real experiments require specialist facilities and controls. The educational task here is analysing models and evidence, not building a high-voltage apparatus. The useful outcome is someone who can read a plasma claim, identify the quantity and boundary, test a proportional relationship and recognise which part still requires measurement or expert judgement.
CHAPTER 16 OF 19
16. Pressure and boundaries connect plasma science with applications
Plasma pressure and magnetic stress compete in many settings. A useful ideal comparison is beta, β = p/(B²/(2μ₀)), where p is the specified plasma pressure and B²/(2μ₀) is magnetic energy density expressed in pressure units. Definitions should state which pressure components are included.
At fixed pressure, doubling magnetic-field magnitude increases magnetic pressure by a factor of four and reduces beta to one quarter. At fixed field, doubling plasma pressure doubles beta. This is another example in which a field changes a system comparison through a square, unlike the inverse dependence in the earlier drift-speed calculation.
Beta alone does not determine stability or performance. Geometry, gradients, currents and transport remain important. Ranking two devices by one parameter can hide the conditions that actually limit their operation. A useful ratio organises a question; it does not replace the full model.
Boundaries matter in technological applications as well. In surface processing, plasma particles and radiation interact with materials, while the near-surface electrical region affects how charged particles arrive. Desired chemical changes must be distinguished from heating, damage and changes in the surface itself.
In space propulsion, momentum transfer, energy input and system efficiency form another coupled problem. In space-weather research, measurements must be connected with evolving fields and flows across large distances. These applications share plasma principles while imposing different practical goals and constraints.
The learning transfer is to carry the mechanism into the application without assuming that success in one setting predicts success in another. Identify the property to control, the boundary where it matters and the observation that would show the desired change.
Choose a quantity with its model attached
| Quantity | Ideal dependency | Main interpretive check |
|---|---|---|
| Electron Debye length | Square root of temperature over density | Shielding model and scale |
| Electron plasma frequency | Square root of electron density | Angular versus ordinary frequency |
| Gyroradius | Perpendicular momentum over charge and field | Particle regime and geometry |
| Alfvén speed | Field over square root of mass density | Fluid-mode assumptions |
| Energy confinement scale | Stored energy over loss power | Boundary and transient terms |
The table summarises selected ideal relations, not universal predictions for every plasma regime.
Is every plasma extremely hot?
No. Plasma components can have different temperatures, and some partly ionised technological plasmas have relatively cool bulk gas. Specify electron, ion and neutral behaviour instead of assigning one universal temperature to the word plasma.
Does quasineutrality mean the electric field is zero?
No. It describes approximate charge balance over an appropriate scale. Fields and currents can remain important, and charge separation becomes significant in some regions and processes. The approximation must fit the question.
Does a magnetic field directly heat a particle?
The magnetic part of the Lorentz force does no work in the ideal single-particle expression. Electric fields and other processes can change particle energy. A real time-dependent electromagnetic system needs its full energy transfer examined.
Are energy confinement time and particle lifetime identical?
No. Energy and particles have different balances and transport routes. A stock divided by a loss rate gives a characteristic scale under specified assumptions, not a statement that every individual particle follows one fixed schedule.
Does fusion gain prove net electricity production?
Only if the quoted metric actually measures that full electrical boundary. Plasma energy gain, target gain and facility electrical balance are different questions. Definitions and auxiliary inputs must remain attached to the result.
Why do both fluid and kinetic models exist?
They retain different information and suit different scales and processes. Fluid models work with averaged quantities and closures; kinetic models describe distributions in phase space. The observable and physical regime determine which description is adequate.
Choose a route by the question you want to answer. These connected guides develop the specialist mechanisms and adjacent subjects; they are companions to this explanation rather than substitutes for checking a particular real-world case.
Build the plasma mechanisms
How Science Works | Plasma Physics — Ionisation, Collective Fields, Waves, Instabilities and Confinement — Continue into the fuller library of collective fields, instabilities and confinement.
How Science Works | Landau Damping — How a Plasma Wave Loses Energy Without Collisions — Follow resonant energy exchange between particles and a wave without ordinary collisional damping.
How Science Works | Debye Screening — How Ions Shorten the Reach of Electric Fields — Compare screening in an electrolyte with the plasma shielding argument, noting the different mobile species and model assumptions.
How Science Works | Why a Spectral Line Has Width, Not Just a Colour — Examine why spectral width requires several possible contributions to be considered.
How Science Works | Electromagnetism — Charges, Fields, Circuits, Induction and Electromagnetic Waves — Build the electric-field, magnetic-field and force foundations.
How Science Works | Fluid Dynamics — Flow, Pressure, Viscosity, Turbulence and Tested Predictions — Compare fluid reasoning with the additional electromagnetic and kinetic structure of plasma.
Connect energy and space settings
How Nuclear Energy Works | Fission, Fusion, Binding Energy and Reactor Systems — Distinguish nuclear reaction energy from plasma and facility energy balances.
How Science Works | Nuclear Physics — Nuclei, Binding Energy, Decay, Reactions and the Evidence Inside Matter — Connect fusion reactions with nuclei, binding energy and decay.
How Science Works | Astrophysics — Stars, Gravity, Radiation, Compact Objects and the Physics of the Universe — Place laboratory mechanisms beside the different conditions found in cosmic systems.
How Science Works | How Falling Matter Forms a Hot Accretion Disc — Follow heating and transport in matter moving around a compact object.
How Science Works | How Magnetars Produce Extreme Magnetic Activity — Explore an extreme magnetic environment while preserving the limits of laboratory comparisons.
Continue across Gold Standard subjects
The Gold Standard of Electromagnetism provides the connected electromagnetism perspective.
The Gold Standard of Fluid Mechanics provides the connected fluid mechanics perspective.
The Gold Standard of Nuclear Physics provides the connected nuclear physics perspective.
The Gold Standard of Statistical Mechanics provides the connected statistical mechanics perspective.
The Gold Standard of Astronomy provides the connected astronomy perspective.
For the wider collection, return to How X Works and How Science Works.
These primary institutional and research sources were checked on 21 October 2026. They support the scientific framing and specialist references; the numerical cases, teaching sequence and interpretation checklists are original educational examples.
Department of Energy: plasma — Ionisation, plasma variety and applications.
Richard Fitzpatrick: Plasma Physics — University of Texas lecture notes on particles, fluid models, waves and instabilities.
University of Texas: Debye shielding — The ideal shielding model and its characteristic scale.
University of Texas: plasma frequency — Collective electron oscillation and the associated timescale.
Department of Energy: fusion energy science — The multidisciplinary relationship between plasma science and controlled fusion.
Return to the source appropriate to the specific measurement, method and context before using a classroom model to make a professional decision. Where the article states a conditional example, its conclusion depends on the assumptions explicitly given with it.
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