pressure is mostly treated as a scalar , however there are cases (outside of General Relativity where the pressure , like the energy density, has to appear as components of a tensor, in order to maintain the covariant character of the theory) mentioned in advanced textbooks and in research papers where pressure is treated as a tensor , such as in the statistical mechanics of inhomogeneous fluids .
Pressure is the force per unit area acting across a surface element inside a fluid, and is the sum of a kinetic contribution due to the motion of molecules across the surface (convective momentum transport) and a configurational contribution caused by intermolecular forces. For a homogeneous fluid the pressure is the same in all directions, it is represented by the scalar quantity pp. In an inhomogeneous fluid it is a second rank tensor p , as the pressure depends on the orientation of the surface and on the direction relative to the surface.There will exist a shear or tangential pressure for an element randomly oriented.
Thus pxyxy gives the force per unit area in the yy-direction on a surface pointing in
the xx-direction.
The pressure tensor p(r) at a point r is defined as the ensemble average of the negative of the stress tensor :
The pressure tensor is also related to the rate of change of the density of linear momentum at r and at time t , J(r,t) :
For certain limiting conditions , one obtains a tensor called the Irving Kirkwood pressure tensor .
So the pressure tensor of a fluid can be regarded as the negative of the equilibrium average of the stress tensor at the microscopic scale .The pressure tensor is thus composed of two parts in a fluid of spherical molecules ( a classical system of interacting particles) , a kinetic part arising from the linear momentum of the molecules , and a configurational part arising from the central intermolecular forces :
The first part is usually well defined , whereas in inhomogeneous fluids the second is not , because there is not an unambiguous way of deciding where in the fluid a certain intermolecular force is acting.
The pressure tensor is also used in the study of macromolecular systems .Pressure is a useful thermodynamic quantity in constant pressure and volume simulations. The pressure for an atomic system is given by :
In the equation above , PP is the pressure , VVthe volume , mjmj the atomic masses , vjvjthe atomic velocities , rjrj the atomic coordinates , and FjFj the forces . The pressure tensor can be expressed as :
An atom representation or a center of mass representation may be used for a molecular system.
It is to note that there is a pressure tensor defined in relation to the Vlasov equationas the particle mass times the covariance matrix of the velocity (equation image below from Wikipedia):
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