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Phase Diagram

Phase Diagram

The phase of a given substance depends on the pressure and temperature. Thus, plots of pressure versus temperature showing the phase in each region provide considerable insights into the thermal properties of substances. Such plots are known as phase diagrams. For instance, in the phase diagram for water (Figure 1), the solid curve boundaries between the phases indicate phase transitions (i.e., temperatures and pressures at which the phases coexist).

Phase Diagrams

Phase Diagrams

A phase diagram combines plots of pressure versus temperature for the liquid-gas, solid-liquid, and solid-gas phase-transition equilibria of a substance. These diagrams indicate the physical states that exist under specific conditions of pressure and temperature and also provide the pressure dependence of the phase-transition temperatures (melting points, sublimation points, boiling points). Regions or areas labeled solid, liquid, and gas represent single phases, while lines or curves represent...

Phase Transitions: Sublimation and Deposition

Phase Transitions: Sublimation and Deposition

Some solids can transition directly into the gaseous state, bypassing the liquid state, via a process known as sublimation. At room temperature and standard pressure, a piece of dry ice (solid CO2) sublimes, appearing to gradually disappear without ever forming any liquid. Snow and ice sublimate at temperatures below the melting point of water, a slow process that may be accelerated by winds and the reduced atmospheric pressures at high altitudes. When solid iodine is warmed, the solid sublimes...

pV-Diagrams

pV-Diagrams

The pV diagram, which is a graph of pressure versus volume of the gas under study, is helpful in describing certain aspects of the substance. When the substance behaves like an ideal gas, the ideal gas equation describes the relationship between its pressure and volume. On a pV diagram, it is common to plot an isotherm, which is a curve showing p as a function of V with the number of molecules and the temperature fixed. Then, for an ideal gas, the product of the pressure of the gas and its...

Phase Transitions: Melting and Freezing

Phase Transitions: Melting and Freezing

Heating a crystalline solid increases the average energy of its atoms, molecules, or ions, and the solid gets hotter. At some point, the added energy becomes large enough to partially overcome the forces holding the molecules or ions of the solid in their fixed positions, and the solid begins the process of transitioning to the liquid state or melting. At this point, the temperature of the solid stops rising, despite the continual input of heat, and it remains constant until all of the solid is...

Phase Changes

Phase Changes

Phase transitions play an important theoretical and practical role in the study of heat flow. In melting or fusion, a solid turns into a liquid; the opposite process is freezing. In evaporation, a liquid turns into a gas; the opposite process is condensation.
A substance melts or freezes at a temperature called its melting point and boils or condenses at its boiling point. These temperatures depend on pressure. High pressure favors the denser form of the substance, so typically, high pressure...

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Related Experiment Video

Updated: Nov 20, 2025

Phase Diagram Characterization Using Magnetic Beads as Liquid Carriers

Phase Diagram Characterization Using Magnetic Beads as Liquid Carriers

Published on: September 4, 2015

Uncomputability of phase diagrams.

Johannes Bausch¹, Toby S Cubitt², James D Watson³

¹CQIF, DAMTP, University of Cambridge, Cambridge, UK. jkrb2@cam.ac.uk.

Nature Communications

|January 20, 2021

Summary

This summary is machine-generated.

Determining the phase diagram of many-body Hamiltonians is generally uncomputable. This study proves undecidability in a positive measure of parameter space, advancing condensed matter physics research.

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Area of Science:

Condensed matter physics
Quantum many-body systems
Computational complexity theory

Background:

Phase diagrams are crucial for understanding material properties and condensed matter systems.
Previous research indicated undecidability in phase diagram computation only on zero-measure sets within parameter space.

Purpose of the Study:

To prove the general uncomputability of determining phase diagrams for many-body Hamiltonians.
To demonstrate that undecidability can exist over a positive measure of a Hamiltonian's parameter space.

Main Methods:

Construction of a continuous one-parameter family of Hamiltonians, H(φ).
Analysis of translationally-invariant Hamiltonians with nearest-neighbor couplings on a 2D spin lattice.

Main Results:

Explicit proof of the uncomputability of determining phase diagrams for the constructed Hamiltonians.
Demonstration that undecidability applies to a set of positive measure in the Hamiltonian parameter space.

Conclusions:

The task of computing phase diagrams for many-body systems is generally undecidable.
This finding extends undecidability results for spectral gaps closer to practical condensed matter problems involving continuous parameters.