Locally accurate matrix product approximation to thermal states

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...

Sigmatropic rearrangements are a class of pericyclic reactions in which a σ bond migrates from one part of a π system to another. These are intramolecular rearrangements where the total number of σ and π bonds remain unchanged.
Sigmatropic shifts are classified based on an order term [i, j ], where i and j indicate the number of atoms across which each end of the σ bond migrates. Below are examples of a [3,3] sigmatropic shift in...

San Francisco's Golden Gate Bridge is exposed to temperatures ranging from -15 °C to 40 °C. At its coldest, the main span of the bridge is 1275 m long. Assuming that the bridge is made entirely of steel, what is the change in its length between these temperatures?
To solve the problem, first, identify the known and unknown quantities. The initial length (L) of the bridge is 1275 m, the coefficient of linear expansion (α) for steel is 12 x 10-6/°C, and the change in...

Heat and temperature are essential concepts for everyone every day. The study of heat and temperature is part of an area of physics known as thermodynamics. It is not always easy to distinguish heat and temperature.
The concept of temperature has evolved from the common concepts of hot and cold. The scientific definition of temperature explains more than just our sense of hot and cold. Temperature is operationally defined as the quantity measured with a thermometer. Furthermore, temperature is...

Thermal RadiationThermal radiation is a type of energy transfer that occurs through electromagnetic waves, especially infrared radiation. Unlike conduction and convection, thermal radiation does not need a medium and can even travel through the vacuum of space. This is how the Sun heats the Earth. All objects emit thermal radiation, and the hotter an object is, the more radiation it gives off.Thermal radiation plays an important role in our daily lives and technology. It helps in heating...

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....