{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Velocity distributions\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/velocity-distributions). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "A gas is described by a distribution of velocities, not a single representative speed. Histograms and summary moments expose different aspects of the same samples. A distribution is a data product with a bin policy and sampling context, not a decorative collection of bars.\n", "\n", "## Velocity distributions in Python\n", "\n", "Read the relationship as a short program. Begin with measured or chosen values, keep only the state needed for the next calculation, apply the stated rule, and inspect a result that can be checked independently." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "rng = np.random.default_rng(42)\n", "mass, k_b, temperature = 4.65e-26, 1.380649e-23, 300.0\n", "components = rng.normal(0, np.sqrt(k_b * temperature / mass), (100_000, 3))\n", "speeds = np.linalg.norm(components, axis=1)\n", "print(\"Mean speed (m/s):\", speeds.mean())\n", "print(\"RMS speed (m/s):\", np.sqrt(np.mean(speeds**2)))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "$$\n", "\\langle v^2\\rangle=\\frac1N\\sum_{i=1}^{N}v_i^2\n", "$$\n", "\n", "\n", "Mean speed, mean-square speed, and most-probable speed are different reducers. Squaring emphasizes faster particles, which is why mean-square speed connects naturally to kinetic energy. A component histogram can be symmetric around zero even while the speed-magnitude histogram has no negative values. Before comparing a simulation with an equilibrium Maxwell–Boltzmann shape, subtract any bulk center-of-mass velocity or explicitly model the drift.\n", "\n", "Bin width trades detail against noise. Very narrow bins can make finite samples look jagged; very wide bins hide structure such as two populations. A sound workflow stores raw samples, fixed edges, count totals, overflow policy, and the sample window. It can then test whether a conclusion—such as a broadened distribution after heating—survives reasonable re-binning rather than depending on a chosen bar width.\n", "\n", "A useful check starts with a prediction: **Increasing equilibrium temperature broadens a random velocity distribution, while a uniform drift shifts component means without necessarily broadening random thermal motion.** Define a bin policy for values on an edge and a test that catches dropped samples. Compare two bin widths over the same raw data, then distinguish a shifted component mean from an increased mean-square speed after bulk motion is removed.\n", "\n", "The explanation above has deliberate limits. Few samples produce noisy histograms, and non-equilibrium systems may not follow an equilibrium distribution at all. Quantum statistics, interactions, mixtures, spatial gradients, driven systems, selection bias, and measurement resolution can change the expected shape. A histogram never proves equilibrium by itself; it needs time, spatial, and model-context checks.\n", "\n", "## Equilibrium component distribution\n", "\n", "For a classical ideal gas at equilibrium with zero bulk flow, one Cartesian velocity\n", "component follows a Gaussian distribution:\n", "\n", "\n", "$$\n", "f(v_x)=\\sqrt{\\frac{m}{2\\pi k_BT}}\\exp\\left(-\\frac{mv_x^2}{2k_BT}\\right)\n", "$$\n", "\n", "\n", "It is symmetric about zero, so $\\langle v_x\\rangle=0$,\n", "while $\\langle v_x^2\\rangle=k_BT/m$.\n", "The same holds independently for the other Cartesian components in an isotropic ideal\n", "gas. Adding a uniform drift shifts component means; subtracting that bulk velocity restores\n", "the thermal distribution.\n", "\n", "## Speed is not a velocity component\n", "\n", "Speed $v=|\\mathbf v|$ is\n", "nonnegative. Combining three independent component distributions with the spherical\n", "velocity-space volume factor gives the Maxwell speed density\n", "\n", "\n", "$$\n", "P(v)=4\\pi\\left(\\frac{m}{2\\pi k_BT}\\right)^{3/2}v^2\\exp\\left(-\\frac{mv^2}{2k_BT}\\right),\\qquad v\\ge0\n", "$$\n", "\n", "\n", "The factor $v^2$\n", "means zero speed is not the most common even though every component Gaussian peaks at\n", "zero. The density is normalized:\n", "\n", "\n", "$$\n", "\\int_0^\\infty P(v)\\,dv=1\n", "$$\n", "\n", "\n", "Three characteristic speeds answer different questions:\n", "\n", "\n", "$$\n", "v_{\\mathrm{mp}}=\\sqrt{\\frac{2k_BT}{m}},\\qquad \\langle v\\rangle=\\sqrt{\\frac{8k_BT}{\\pi m}},\\qquad v_{\\mathrm{rms}}=\\sqrt{\\frac{3k_BT}{m}}\n", "$$\n", "\n", "\n", "They obey $v_{\\mathrm{mp}}<\\langle v\\rangle= edges[0]) & (speeds < edges[-1])\n", "counts, _ = np.histogram(speeds[inside], bins=edges)\n", "underflow = np.count_nonzero(speeds < edges[0])\n", "overflow = np.count_nonzero(speeds >= edges[-1])\n", "assert counts.sum() + underflow + overflow == len(speeds)\n", "density = counts / (len(speeds) * np.diff(edges))\n", "print(\"Probability represented inside bins:\", np.sum(density * np.diff(edges)))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "Half-open bins include the left edge and exclude the right, except for a separately stated\n", "final-edge rule. This prevents a sample on an edge from being counted twice or dropped.\n", "Compute moments from raw samples, not from rounded bar centers.\n", "\n", "## Temperature and mass change the shape\n", "\n", "Raising temperature broadens the distribution and moves its speed peak rightward. Every\n", "characteristic speed scales as $\\sqrt T$.\n", "Increasing mass at fixed temperature narrows the distribution and moves the peak left.\n", "\n", "A mixture of different molecular masses does not follow one Maxwell speed curve with an\n", "average mass. Each species has its own distribution at the common equilibrium\n", "temperature. Plot and fit them separately before combining counts.\n", "\n", "Two populations at different temperatures can produce a broad or double-structured\n", "histogram that shares moments with a single equilibrium curve. Matching only\n", "$\\langle v^2\\rangle$ does not prove\n", "equilibrium.\n", "\n", "## Sampling and uncertainty\n", "\n", "Particles from one snapshot are correlated when interactions are important, and snapshots\n", "close in time are correlated because trajectories persist. Estimate uncertainty using\n", "independent runs or blocks longer than the relevant correlation time. Narrow bins have\n", "larger count noise; wide bins hide structure.\n", "\n", "Compare empirical cumulative probabilities as well as bar heights, because a cumulative\n", "distribution avoids arbitrary bin edges. Quantile checks can expose incorrect tails even\n", "when the center looks plausible.\n", "\n", "## Verification\n", "\n", "Generate independent Gaussian velocity components with variance\n", "$k_BT/m$. Check near-zero\n", "component means, equal directional variances, and the predicted characteristic speeds.\n", "Add a constant drift, verify shifted component means, then remove the measured bulk\n", "velocity and recover the original thermal moments.\n", "\n", "Re-bin the same raw samples several ways. Counts and raw moments must remain fixed even\n", "though bar appearance changes. Verify histogram area, overflow accounting, nonnegative\n", "speeds, and the ordering of characteristic speeds. Increase sample count and require\n", "statistical disagreement with the analytic distribution to shrink within uncertainty.\n", "\n", "The Maxwell–Boltzmann form assumes a dilute classical equilibrium gas. Quantum degeneracy,\n", "strong interactions, relativistic speeds, external driving, gradients, and selection\n", "bias require other distributions. A histogram is evidence to compare with a model, not a\n", "standalone proof of equilibrium.\n", "\n", "## A worked Python notebook\n", "\n", "Calculate the root-mean-square speed of molecules with mass 4.65 times ten to the minus 26 kilograms. The question for this calculation is precise: how does **root-mean-square molecular speed**\n", "change when we change **temperature**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "The gas is classical and in thermal equilibrium; RMS speed differs from the mean speed. Keep that sentence beside the results. A formula can return a number outside\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "100 to 600 K. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for temperature, expressed in K. The output is\n", "root-mean-square molecular speed, expressed in m/s. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = 29.8452892484322 \\sqrt{q}\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 300 K, the expression gives 516.936 m/s. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"sqrt(3*1.380649e-23*q/4.65e-26)\", locals={\"q\": q})\n", "reference_input = 300.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"K\")\n", "print(\"Predicted root-mean-square molecular speed:\", reference_output, \"m/s\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "298.453 m/s. The midpoint, at 350 K, gives\n", "558.354 m/s. The upper endpoint gives 731.057 m/s.\n", "Read these together with the assumptions: they describe different choices of temperature\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([100.0, 350.0, 600.0])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(f\"{input_value:.6g} K -> {output_value:.6g} m/s\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows temperature; the vertical axis shows root-mean-square molecular speed. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "input_grid = np.linspace(100.0, 600.0, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Temperature (K)\")\n", "ax.set_ylabel(\"Root-mean-square molecular speed (m/s)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "root-mean-square molecular speed change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 300 K is\n", "0.861559 in those ratio units. Locally, the output increases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = \\frac{14.9226446242161}{\\sqrt{q}}\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 302.5\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"m/s\")\n", "print(\"Full expression:\", full_prediction, \"m/s\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"m/s\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "Here the changed input is 302.5 K. The tangent estimate is\n", "519.089 m/s, while the full expression gives\n", "519.085 m/s. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "input_span = 500.0\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "temperature inside the stated interval, predict whether root-mean-square molecular speed will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "Finally, return to the physical limit stated at the start: The gas is classical and in thermal equilibrium; RMS speed differs from the mean speed. Describe one change\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of m/s:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(600.0) - evaluate(100.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(100.0, 600.0, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "- [OpenStax, *University Physics Volume 2*, Distribution of Molecular Speeds](https://openstax.org/books/university-physics-volume-2/pages/2-5-distribution-of-molecular-speeds).\n", "- D. V. Schroeder, *An Introduction to Thermal Physics*, velocity statistics and sampling.\n", "- [OpenStax, *University Physics Volume 2*, Chapter 2: The Kinetic Theory of Gases](https://openstax.org/books/university-physics-volume-2/pages/2-introduction).\n", "- [MIT 8.01SC, Chapter 29: Kinetic Theory of Gases](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter29/).\n", "- [Benjamin Crowell, *Light and Matter*, pressure and temperature statistics](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, “Kinetic-Molecular Theory”](https://physics.info/kmt/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "velocity-distributions.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/velocity-distributions", "source_sha256": "92f8f101baebfd9f0c82886e45ff33243af02d25fb37ed6b990b81315458afff" } }, "nbformat": 4, "nbformat_minor": 5 }