TRAINING PROGRAMME IN
COLLABORATIVE MATHEMATICAL RESEARCH
This fourth call is addressed only to doctoral students.
in this call can be consulted through the following
Complex Systems. Website. Projects.
Industrial Mathematics. Website. Projects.
Mathematical Epidemiology. Website. Projects.
How to apply
To apply for a position, you must send to email@example.com an email with the following
information: name, personal address, email, phone number, and as attached documents
(CV, academic records, incorporating, if possible, the computation of the average grade
following the same scheme as for the FI-DGR 2014 call, and a motivation letter. A
minimum of one and a maximum of three hosting CRM research groups have to be chosen,
indicating preferences in the motivation letter, if any.) *
* (please name all attached files including your family name for easy identification).
The deadline to submit an application is April 17, 2015. Applicants should be available for
interviews (possibly via Skype) from April 23 onwards.
Related calls: Industrial Doctorate at CRM
Nanoscale heat transfer
The classical theory of heat transport is based on Fourier’s law, which states that the
flow of heat is proportional to the temperature gradient. This assumption leads to the
classical form of the heat equation, which has been successfully used to model the
temperature in materials for over 200 years. However, as technology advances,
situations arise where the standard heat equation is no longer accurate and certain
accepted properties turn out to be invalid.
Experimental data and simulations have demonstrated that at the nanoscale heat does
not necessarily flow in the classical manner. For example, experiments of laser heating
of ultrathin layers or simulations of heat transport in solids using molecular dynamics
show dramatic discrepancies with respect to classical laws. The unpredictable behaviour
makes the design stage of future nanoscale devices very difficult. Understanding heat
transport at this scale and proposing modified versions of the classical equations (that
prove to be valid) is a key point in order to ease the design of these future devices.
Various attempts have been made to develop an accurate mathematical model for heat
flow, perhaps two of the most well-known are the Maxwell-Cattaneo and the Guyer-
Krumhansl equations. The first introduces a relaxation time into the heat flow expression
that has the effect of changing the governing equation to a form of wave equation, which
then exhibits significantly different behaviour to the standard heat equation. The second
introduces nonlocal effects that incorporate interesting new phenomena such as heat
In this project we intend to examine these novel forms of heat equation, and study their
validity at the nanoscale. We will investigate:
- The effect of factors such as the relaxation time, nonlocalities, size dependent
material properties (which results in a higher order equation).
- How the new forms of heat equation affect the heat flow and at what length-scales
they are valid, i.e. when is the classical heat equation sufficiently accurate?
- The effect of boundary and initial conditions: indeed what are appropriate conditions
at the nano-scale?
The results of the mathematical study will be linked to experimental observations. Do
the results confirm experimental findings? Is any new behaviour predicted by the work
and can this be used to guide future nano-scale experiments or the design of nano-
For further reading see:
Wikipedia, Relativistic heat conduction.
F.X. Álvarez, D. Jou. Boundary Conditions and Evolution of Ballistic Heat Transport. J.
Heat Transfer 132(1), 2010, doi:10.1115/1.3156785
A. Sellitto, F.X. Álvarez, D. Jou. Second law of thermodynamics and phonon-boundary
conditions in nanowires. J. Appl. Phys 107(6), 2010, doi: 10.1063/1.3309477
Supervisors: Prof. Tim Myers (CRM), Prof. Xavier Álvarez (Física Estadística, UAB)