Random Logo Collection

108 standalone TikZ images, mostly about physics and machine learning.

Built by Janosh Riebesell.

  TikZ code on GitHub.     MIT licensed. Free to reuse.

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Autoencoder

machine learningneural networks

Autoencoder

Variational autoencoder architecture. Made with https://github.com/battlesnake/neural.

Aviary

machine learningneural networksmaterials informatics

Aviary

Aviary logo: a repo of machine learning models for representation learning in materials discovery from stoichiometry and symmetries via graph convolutions. For https://github.com/CompRhys/aviary.

Bloch Sphere

physicsquantum mechanics

Bloch Sphere

A Bloch sphere of radius |a| = 1 contains all possible states of a two-state quantum system (qubit). Each Bloch vector fully determines a spin-1/2 density matrix. Used in Exercise Sheet 10 of Statistical Physics by Manfred Salmhofer (2016), available at https://janosh.dev/physics/statistical-physics.

Bose Einstein Distribution

physicsstatistical mechanics

Bose Einstein Distribution

Illustrating the change in the real part of the Bose-Einstein distribution, i.e. the average occupancy of the ground state of a bosonic system, from doubling the temperature. Pulled from arxiv:1712.09863.

Bose Einstein Distribution 3d

physicsstatistical mechanicsMatsubara

Bose Einstein Distribution 3d

Surface plot of the Bose-Einstein distribution over the complex plane. This plot shows an important feature of the Matsubara formalism developed for QFT at non-zero temperature. It's a method to calculate expectation values of operators in a canonical ensemble evolved by an imaginary time. In momentum space, this leads to the replacement of continuous frequencies by discrete imaginary (Matsubara) frequencies. Away from the imaginary axis, the distribution becomes approximately flat, particularly at sufficiently low temperatures. Pulled from arxiv:1712.09863.

Branch and Bound

optimizationalgorithm

Branch and Bound

Illustration of a linear-programming (LP)-based branch and bound algorithm. Not knowing how to solve a mixed-integer programming (MIP) problem directly, it first removes all integrality constraints. This results in a solvable LP called the linear-programming relaxation of the original MIP. The algorithm then picks some variable x restricted to be integer, but whose value in the LP relaxation is fractional. Suppose its value in the LP relaxation is x = 0.7. It then excludes this value by imposing the restrictions x ≤ 0 and x ≥ 1, thereby creating two new MIPs. By applying this recursively step and exploring each resulting bifurcation, the globally optimal solution satisfying all constraints can be found.

Branch Cuts 1

physicsquantum field theory

Branch Cuts 1

Propagator branch cuts, i.e. a continuum of singularities, along the real frequency axis extending from ±p\pm |\vec p| out to ±\pm \infty. Pulled from arxiv:1712.09863.

Branch Cuts 2

physicsquantum field theoryMatsubara

Branch Cuts 2

Branch cuts approaching the real q0q_0-axis with decreasing kk where they merge at k=0k = 0. kk denotes the energy cutoff of a scalar field theory.

Change of Variables

mathprobabilitystatistics

Change of Variables

Simple 2d example illustrating the role of the Jacobian determinant in the change of variables formula. Inspired by Ari Seff in https://youtu.be/i7LjDvsLWCg?t=250.

Closed String Topologies

physicsstring theory

Closed String Topologies

When calculating scattering amplitudes via the path integral, we must sum over all possible world-sheet topologies. To characterize the types of world-sheets that have to be considered at each level of its perturbative expansion, string theory makes use of the following theorem:

Every compact, connected, oriented two-dimensional manifold is topologically equivalent to a sphere with gg handles (gg for genus) and bb boundaries. A topological invariant of two-dimensional oriented surfaces is the Euler characteristic χ=22gb\chi = 2 - 2g - b.

What this boils down to is that we can obtain the topological characteristics of higher and higher loop-level world-sheet topologies by successively increasing in one-step increments the number of handles gg in case of the closed string and the number of boundaries bb for the open sector. This gives the topologies in this figure for the vacuum diagram of the closed sector up to one-loop level. For the open sector, see open string topologies.

Complex Sign Function

physicsquantum field theoryMatsubara

Complex Sign Function

3d surface plot of the complex sign function s(p0)=\sign(p0p0)s(p_0) = \sign(\Re p_0 \, \Im p_0) over the complex plane. Used in the Matsubara summation of thermal quantum field theory to split contour integrals in the complex plane into two parts, the first being branch-cut free and the second evident branch cut structure.

Concave Functions

mathphysicsstatistical physics

Concave Functions

xln(x)x \ln(x) is convex and x1x - 1 is its tangent, so xln(x)x1xx \ln(x) \geq x - 1 \enskip \forall x.

Conv2d

machine learningneural networkscomputer vision

Conv2d

A two-dimensional convolution operator slides the kernel matrix across the target image and records elementwise products. Makes heavy use of the matrix environment in TikZ.

Convex Functions

math

Convex Functions

xx and xlog(x)-x\log(x) are concave functions. Since ln(p)\ln(p) decomposes into sums of these two components, it too must be concave. Any extremum of a concave function is a maximum. This fact is used in statistical physics to find the equilibrium distribution of many-particle systems. See problem 2 on this exercise sheet.

Critical Temperature

physicsthermodynamics

Critical Temperature

Cylinder to Plane

physicsstring theorytopology

Cylinder to Plane

String theory: Primary fields and radial quantization A time-ordered product of fields on the cylinder maps to a radially ordered product in the complex plane. This graphic visualizes how different times on the cylinder correspond to different times on the plane.

Detailed Balance

physicskinematics

Detailed Balance

DFT Choices

physicssolid state physicsdensity functional theory

DFT Choices

Visualization of the many choices involved in a Kohn-Sham calculation. It can be non-relativistic based on the classical Schrödinger equation or fully relativistic based on the Dirac equation which includes spin-orbit coupling. The near-core electrons can be modeled explicitly in an all-electron calculation or, much more commonly, incorporated along with the nucleus into an effective pseudo-potential. The Hartree energy can be obtained by integrating the charge density or by solving the Poisson equation. The exchange-correlation potential can be treated with a huge library of density functionals, most commonly LDA and GGA. The wave functions can be computed on a numerical mesh and expanded in one of many possible basis sets, e.g. plane waves, APW, or PAW. Inspired by arxiv:cond-mat/0211443.

Diagrams

physicsquantum field theory

Diagrams

Disk to Plane

physicsstring theorytopology

Disk to Plane

Distributions

physicsstatistical mechanics

Distributions

Divergence

math

Divergence

Dropout

machine learningneural networksregularization

Dropout

Illustration of applying dropout with a rate of p=0.5p = 0.5 to a multilayer perceptron.

Ergodic

physicsstatistical mechanics

Ergodic

Fermi

physicsstatistical mechanics

Fermi

Feynman 1

physicsquantum field theory

Feynman 1

Feynman 2

physicsquantum field theory

Feynman 2

Feynman 3

physicsquantum field theory

Feynman 3

Feynman 4

physicsquantum field theory

Feynman 4

Feynman Diagram Propagator Loop

physicsquantum field theory

Feynman Diagram Propagator Loop

This Feynman diagram contains two propagators forming a loop carrying the external energy q0q_0. m1,2m_{1,2} denote the masses of the propagators and γ1,2\gamma_{1,2} their decay width which, for an expansion in Minkowski space, are non-zero only around a real and positive on-shell frequency p0>0p_0 > 0.

Fluctuations

physicsstatistical mechanics

Fluctuations

Number fluctuations of the occupation probability〈nk〉of a single mode k in an ideal Bose and Fermi gas in the grand canonical ensemble. Used in Exercise Sheet 11 of Statistical Physics by Manfred Salmhofer (2016), available at https://janosh.dev/physics/statistical-physics.

Four Vs of Data

machine learningmaterials informaticsinfo-graphics

Four Vs of Data

The state of the 4 Vs of data as they apply to materials informatics. Inspired by fig. 1 in https://api.semanticscholar.org/CorpusID:137734316.

GAN

machine learningneural networksgenerative modeling

GAN

Generative adversarial network (GAN) architecture. A GAN has two parts. The discriminator DD acts as a classifier that learns to distinguish fake data produced by the generator GG from real data. GG incurs a penalty when DD detects implausible results. This signal is backpropagated through the generator weights such that GG learns to produce more realistic samples over time, eventually fooling the discriminator if training succeeds.

Geometric Bayes

Bayesianprobabilitymath

Geometric Bayes

3blue1brown-inspired geometric visualization of Bayes theorem https://youtu.be/HZGCoVF3YvM.

Graph Isomorphism

mathsymmetrygraphs

Graph Isomorphism

Graphs can look differently but be identical. The only thing that matters are which nodes are connected to which other nodes. If there exists an edge-preserving bijection between two graphs, in other words if there exists a function that maps nodes from one graph onto those of another such that the set of connections for each node remain identical, the two graph are said to be isomorphic.

Gravitons

physicsquantum field theory

Gravitons

Harmonic Oscillator Energy vs Angular Frequency

physics

Harmonic Oscillator Energy vs Angular Frequency

harmonic oscillator energy vs angular frequency omega

Harmonic Oscillator Energy vs inverse Temperature

physicsthermodynamics

Harmonic Oscillator Energy vs inverse Temperature

Harmonic oscillator energy vs inverse temperature beta = 1/(kB T)

HEA

solid state physicsentropycatalysis

HEA

Cartoon of the AlCoCrFeNi high entropy alloy (HEA) with body-centered cubic (BCC) lattice. HEAs also come in face-centered cubic (FCC) and (less frequently) hexagonal close packing (HCP). BCC HEAs typically have high yield strength and low ductility, vice versa for FCC HEAs. Wiki article.

Heatmap

info-graphics

Heatmap

Posted as an answer to https://tex.stackexchange.com/q/44868.

Higgs Potential

physicsquantum field theorysymmetry

Higgs Potential

The Higgs mechanism plays a vital role in the Standard Model for explaining how gauge bosons obtain mass. The Standard Model would otherwise predict these particles to be massless. Through interactions with the Higgs field that permeates all space and whose elevated potential at zero field leads to spontaneous symmetry breaking (SSB), gauge bosons also experience symmetry breaking, causing them to acquire mass. See https://tikz.netlify.app/maxican-hat for a very similar image.

Isotherms

physicsthermodynamicsstatistical mechanics

Isotherms

Jensens Inequality

math

Jensens Inequality

k-Space

physicssolid state physics

k-Space

Kohn Sham Cycle

physicsquantum mechanicsdensity functional theory

Kohn Sham Cycle

Loop

physicsquantum field theoryrenormalization

Loop

Loops

physicsquantum field theoryrenormalization

Loops

M-Theory

physicsstring theory

M-Theory

MADE

machine learninggenerative modelingprobabilitystatistics

MADE

TikZ-reproduction of fig. 1 from the paper MADE: Masked Autoencoder for Distribution Estimation (arxiv:1502.03509).

MAF

machine learninggenerative modelingprobabilitystatistics

MAF

Illustration of the slow (sequential) forward pass of a Masked Autoregressive Flow (MAF) layer as introduced in arxiv:1705.07057. Inspired by https://blog.evjang.com/2018/01/nf2.html.

Materials Informatics

machine learninginfo-graphics

Materials Informatics

Structure-based materials informatics workflow. Inspired by fig. 1 in https://doi.org/10.1016/j.cpc.2019.106949.

Materials Informatics Challenges

machine learningsolid state physicsinfo-graphics

Materials Informatics Challenges

Inspired by https://tex.stackexchange.com/a/387466.

Matsubara Contour 1

physicsquantum field theoryMatsubara

Matsubara Contour 1

Complex contour plot illustrating a Matsubara summation. Used in thermal quantum field theory to compute Feynman diagrams at non-zero temperature. CC surrounds the imaginary p0p_0-axis counterclockwise but excludes poles of (p02+x2)1(-p_0^2 + x^2)^{-1}.

Matsubara Contour 2

physicsquantum field theoryMatsubara

Matsubara Contour 2

Deformation of contour CC in Matsubara contour 1 into a circle followed by taking the radius to infinity. This will enclose the poles of (p02+x2)1(-p_0^2 + x^2)^{-1} scattered throughout the complex plane. Their contribution is removed again by enclosing them in clockwise contours.

Matsubara Contour 3

physicsquantum field theoryMatsubara

Matsubara Contour 3

Matsubara Contour 4

physicsquantum field theoryMatsubara

Matsubara Contour 4

Matsubara Contour 5

physicsquantum field theoryMatsubara

Matsubara Contour 5

Contour Deformation

physicsquantum field theoryMatsubara

Contour Deformation

Maxwell Boltzmann Dist

physicsstatistical mechanicsthermodynamics

Maxwell Boltzmann Dist

The Maxwell-Boltzmann distribution plotted at different temperatures reveals that the most probable velocity vpv_p of ideal gas particles scales with the square root of temperature.

Mexican Hat

physicsquantum field theorysymmetry

Mexican Hat

The Mexican hat potential exhibits spontaneous symmetry breaking (SSB), a process by which a physical system starting in a symmetric state spontaneously enters and remains in an asymmetric state. This usually applies to systems whose equations of motion obey a set of symmetries while the lowest-energy state(s) do(es) not. When the system assumes one of its ground states, its symmetry is broken even though the Lagrangian as a whole retains it. See https://tikz.netlify.app/higgs-potential for a very similar image.

In this image, the system starts out in the naive O(N)O(N)-invariant vacuum (blue dot) but quantum fluctuations quickly push it into the real vacuum (red dot) where O(N)O(N) is broken down to O(N1)O(N-1).

MOSFET

physicselectronicssolid state physics

MOSFET

The metal-oxide-semiconductor field-effect transistor, or MOSFET for short, is the most frequently manufactured device in history, with close to 10^23 MOSFETs produced since 1960. As the basic building block of almost all modern electronics, it revolutionized the world economy and triggered our ascent into the information age.

Mphil Gantt

info-graphics

Mphil Gantt

Useless Gantt chart for my Scientific Computing MPhil project at Cambridge University titled "Data-Driven Risk-Conscious Thermoelectric Materials Discovery". See https://github.com/janosh/thermo.

NF Coupling Layer

machine learninggenerative modelingprobabilitystatistics

NF Coupling Layer

Simple 2d example illustrating the role of the Jacobian determinant in the change of variables formula. Inspired by Ari Seff in https://youtu.be/i7LjDvsLWCg?t=250.

Normalizing Flow

machine learninggenerative modelingprobability

Normalizing Flow

A chain of bijections f=f1fkf = f_1 \circ \dots \circ f_k constituting a normalizing flow which step by step transforms a simple distribution p0(z0)p_0(\vec z_0) into a complex one pk(zk)p_k(\vec z_k). The bijections are trained to fit pk(zk)p_k(\vec z_k) to some target distribution px(x)p_x(\vec x). Inspired by Lilian Weng.

One Point

physicsquantum field theoryrenormalization

One Point

This Feynman diagram corresponds to the integrand in this expression for tΓk,a(1)(q)=12i,jk,lNp1,p2p3,p4tRk,ij(p1,p2)Γk,jk(2)(p2,p3)+Rk,jk(p2,p3)Γk,akl(3)(q,p3,p4)Γk,li(2)(p4,p1)+Rk,li(p4,p1).\displaystyle \partial_t \Gamma_{k,a}^{(1)}(q) = -\frac{1}{2} \sum_{\substack{i,j\\k,l}}^N \int_{\substack{p_1,p_2\\p_3^\prime,p_4^\prime}} \frac{\partial_t R_{k,ij}(p_1,p_2)}{\Gamma_{k,jk}^{(2)}(p_2,p_3) + R_{k,jk}(p_2,p_3)} \, \frac{\Gamma_{k,akl}^{(3)}(q,p_3,p_4)}{\Gamma_{k,li}^{(2)}(p_4,p_1) + R_{k,li}(p_4,p_1)}.

Open String Topologies

physicsstring theorytopology

Open String Topologies

When calculating scattering amplitudes via the path integral, we must sum over all possible world-sheet topologies. To characterize the types of world-sheets that have to be considered at each level of its perturbative expansion, string theory makes use of the following theorem:

Every compact, connected, oriented two-dimensional manifold is topologically equivalent to a sphere with gg handles (gg for genus) and bb boundaries. A topological invariant of two-dimensional oriented surfaces is the Euler characteristic χ=22gb\chi = 2 - 2g - b.

What this boils down to is that we can obtain the topological characteristics of higher and higher loop-level world-sheet topologies by successively increasing in one-step increments the number of handles gg in case of the closed string and the number of boundaries bb for the open sector. This gives the topologies in this figure for the vacuum diagram of the closed sector up to one-loop level. For the closed sector, see closed string topologies.

Operator Orderings

physicsquantum field theoryrenormalization

Operator Orderings

Organic Molecule

chemistry

Organic Molecule

Beta-methylamino-DL-alanine - C4H10N2O2. Coordinates found on https://tex.stackexchange.com/q/453740.

Otto Cycle

physicsthermodynamics

Otto Cycle

The Otto cycle is an idealized thermodynamic cycle of a typical spark ignition piston engine. It is the thermodynamic cycle most commonly found in automobile engines.

Periodic Table

physicschemistry

Periodic Table

The periodic table of elements, a graphic formulation of the periodic dependence of elemental properties on their atomic numbers. The table is divided into four roughly rectangular areas called blocks. Rows are called periods, columns are groups. Elements from the same group show similar chemical characteristics. Trends run through the periodic table, with nonmetallic character (keeping their own electrons) increasing from left to right across a period, and from down to up across a group, and metallic character (surrendering electrons to other atoms) increasing in the opposite direction.

PHD Gantt

info-graphics

PHD Gantt

Physics

physicsinfo-graphics

Physics

Plane to Torus

physicsstring theory

Plane to Torus

Plate Capacitor

physicselectrodynamicselectronics

Plate Capacitor

Parallel plate capacitor with dipolar polarization.

Poles

physicsquantum field theoryMatsubara

Poles

Potential Triangle

physicsthermodynamicsstatistical physicsentropy

Potential Triangle

Propagator Fluctuations

physicsquantum field theory

Propagator Fluctuations

Propagators

physicsquantum field theoryMatsubara

Propagators

Cost vs Accuracy in Quantum Mechanics Simulations

physicsquantum mechanicssolid state physicsdensity functional theory

Cost vs Accuracy in Quantum Mechanics Simulations

Cost vs accuracy trade-off for different quantum mechanics approximations. NN denotes the system size, usually the number of electrons. Source: Frank Noe.

Random Forest

machine learningensemblesclassificationregression

Random Forest

Diagram of the random forest (RF) algorithm (Breiman 2001). RFs are ensembles model consisting of binary decision trees that predicts the mode of individual tree predictions in classification or the mean in regression. Every node in a decision tree is a condition on a single feature, chosen to split the dataset into two so that similar samples end up in the same set. RFs are inspectable, invariant to scaling and other feature transformations, robust to inclusion of irrelevant features and can estimate feature importance via mean decrease in impurity (MDI).

Regular vs Bayes NN

machine learningneural networksstatisticsprobability

Regular vs Bayes NN

Relation Space

info-graphicsmarine bio-chemistry

Relation Space

RNVP

machine learningneural networksgenerative modelingnormalizing flows

RNVP

Illustration of the real-valued non-volume preserving (RNVP) affine coupling layer as introduced in arxiv:1605.08803. Inspired by https://blog.evjang.com/2018/01/nf2.html.

Roost Update

machine learninggraphsneural networkssolid state physics

Roost Update

Representation Learning from Stoichiometry (Roost). Graph representation and update mechanism for the composition La2CuO4. Essentially depicting a graph convolution. arxiv:1910.00617

Sabatier Principle

physicschemistry

Sabatier Principle

Illustration of the Sabatier principle in heterogeneous catalysis. Inspired by From the Sabatier principle to a predictive theory of transition-metal heterogeneous catalysis.

Saddle

mathconvexity

Saddle

Sbs Aktionen

info-graphics

Sbs Aktionen

Operations schematic for the German student initiative "Studenten bilden Schüler eV". See https://studenten-bilden-schueler.de.

Seebeck Effect

physicssolid state physicselectrodynamics

Seebeck Effect

Self Attention

machine learninggraphs

Self Attention

Illustrating the attention mechanism from arxiv:1706.03762.

Shell

physicsstatistical mechanics

Shell

Sign Plane

physicsquantum field theoryMatsubara

Sign Plane

Sign function in the complex plane.

Single-head attention

machine learningattention mechanismattention is all you needtransformer

Single-head attention

Flow diagram of single-head attention illustrating the equation Attention(Q,K,V)=softmaxrow(QKd)V\displaystyle \mathrm{Attention}(Q, K, V) = \mathrm{softmax}_\text{row} \left( \frac{Q K^\top}{\sqrt{d}} \right) V with border colors to indicate tensor dimensions.

Skip Connection

machine learningneural networks

Skip Connection

Illustration of skip connection in a residual block. Inspired by the ResNet paper. arxiv:1512.03385

Spontaneous Magnetization

physicssolid state physicsmagnetism

Spontaneous Magnetization

Spontaneous magnetization mm in an Ising ferromagnet appears below the critical temperature TcT_c. The derivative of mm diverges in the limit TTcT \to T_c^-.

Tanh

physicssolid state physicsmagnetism

Tanh

Plot of the hyperbolic tangent. tanh(x) goes to +/- 1 for x to +- infinity while it is approximately linear for abs(x) < 1. Appears in many places in physics, e.g. in the magnetization of an ideal paramagnet of independent spins.

Theory Space

physicsquantum field theoryrenormalization

Theory Space

Tori

physicsstring theorytopology

Tori

Torus

physicsstring theorytopology

Torus

Torus Fundamental Domain

physicsstring theory

Torus Fundamental Domain

Transformations

physicsthermodynamicsstatistical mechanics

Transformations

Equivalence of thermodynamic ensembles through Laplace and Legendre transforms.

Two Point

physicsquantum field theoryrenormalization

Two Point

Two-point propagator flow

Two Point No Cutoff

physicsquantum field theoryrenormalization

Two Point No Cutoff

Two-point propagator flow without cutoff derivative

Unregularized Propagator Diagrams

physicsquantum field theoryrenormalization

Unregularized Propagator Diagrams

VAE

machine learninggenerative modelingBayesianneural networksprobability

VAE

Variational autoencoder architecture. The earliest type of generative machine learning model. Inspired by https://towardsdatascience.com/intuitively-understanding-variational-autoencoders-1bfe67eb5daf.

Wall

physicsthermodynamicsundergradexercise

Wall

Exercise illustration: Compute the pressure of an ideal gas in three dimensions upon a wall at x=0x = 0 that attracts molecules at large distance and repels them at smaller distance. Let the force be given by the potential U(x)=Aeαx+Be2αxU(x) = -A \, e^{-\alpha x} + B \, e^{-2 \alpha x}, with A,B>0A,B > 0.

Wetterich Equation

physicsparticle physicsquantum physicsquantum field theoryrenormalization

Wetterich Equation

The Wetterich eqn. is a non-linear functional integro-differential equation of one-loop structure that determines the scale-dependence of the flowing action Γk\Gamma_k in terms of fluctuations of the fully-dressed regularized propagator [Γk(2)+Rk]1[\Gamma_k^{(2)} + R_k]^{-1}. It admits a simple diagrammatic representation as a one-loop equation as shown in this diagram.

Wyckoff Positions

physicssolid state physicssymmetriesgroup theory

Wyckoff Positions

2D toy crystal with three occupied Wyckoff positions. The shaded areas illustrate the region of the unit cell the relevant atoms are constrained to lie in by specifying an anonymized Wyckoff position for that atom. Reproduction of fig. 2 from "Wyckoff Set Regression for Materials Discovery" by Rhys Goodall, inspired by PyXtal docs.

zT vs n

physicssolid state physicsthermodynamics

zT vs n

Thermoelectric figure of merit zTzT vs carrier concentration nn for Bi2Te3 based on empirical data in αlnσ\alpha - \ln \sigma plot as a thermoelectric material performance indicator. Tuning nn for optimal zTzT involves a compromise between thermal conductivity κ\kappa, Seebeck coefficient SS and electrical conductivity σ\sigma. Increasing the electrical conductivity σ\sigma not only produces an increase in the electronic thermal conductivity κel\kappa_\text{el} but also usually decreases the Seebeck coefficient SS. This makes optimal zTzT difficult to achieve. Plot scales are κ[W/mK][0,10]\kappa [W / m K] \in [0,10], S[mV][0,500]S [mV] \in [0,500], σ[1/(Ωcm)][0,5000]\sigma [1/(\Omega cm)] \in [0,5000].