• magnetic backgrounds
• Lattice QCD simulations
• Jarzynski theorem

The QGP (Quark Gluon Plasma) is thought to have lled the Universe in its early stages. For this reason the study of the strong interacting matter in a magnetic background has a relevant interest. Indeed cosmological models suggest that strong magnetic elds (√eB ∼ 2 GeV) could be reproduced during the electroweak phase transition of the early universe, and consequently might have an impact on subsequent e ects where strong interaction were involved. Large magnetic elds (√eB ∼ 1 MeV) are also present in dense neutron stars called magnetars. Furthermore, in a noncentral heavy ion collision , the spectators could generate magnetic elds which reaches up to √eB ∼ 0.5 GeV. Lattice QCD simulations are convenient tools for studying magnetic properties of this ”material”, even if the usual lattice setup leads to some technical issues due to toroidal geometry, which imposes the quantization of the magnetic background. Consequently, evaluating free energy derivatives respect to the magnetic eld becomes a tricky task, furthermore, evaluating free energy di erences has always been an awkward goal in statistical mechanics.
The proposal of this thesis is to adopt and study a new (for LQCD simulations) free energy measure- ment technique, which involves the usage of the Jarzynski’s relation [1] [2], reproducing/improving measurements done in precedent studies [3]. The identity relates the average of the exponentiated work W executed on a system, switching an Hamiltonian’s parameter λ from an initial value to a nal one, to the exponetiated free energy di erence ∆F related to such process:
e−βW = e−β(F(λfin)−F(λin)).
At rst, the ”physical” content of such relation will be discussed, proving its bond with the second law of Thermodynamics. Furthermore, looking at Jarzynski’s theorem as a consequence (and viceversa) of the validity of the Entropy production uctuation theorem [4], the ”pathological” issues and properties of an algorithm based on this relation can be discussed in a more accurate way, proving that good free energy estimates are a consequence of a good sampling of a precise region of the work distributions.
In [3] it was found that, up to √eB ∼ 0.1 GeV, the response of the material seems to be linear in B, within errors, with a paramagnetic behavior in almost the whole range of temperatures studied. In [5], using HRG (Hadron Resonance Gas) computations, a weak diamagnetism was predicted for low temperature (T ≤ 100 MeV) due to the dominant contribution from pions (see discussion in Appendix B of [6]). Such diamagnetic behavior is not proven by any numerical investigation yet. Only suggestions of weak diamagnetism have been detected [3], [6]. In this thesis the free energy variation due to the magnetic background around T ∼ 90 MeV will be calculated, using the Non equilibrium work relation , looking for a possible diamagnetic behavior. Thus, we tried to obtain the most precise free energy measurements , utilizing the bond between the Jarzynski’s relation and the Entropy production uctuation theorem. At the same time the e ciency of the technique exposed will be compared with other methods, and systematic error sources will be discussed. Then, possible future applications,where the technique is supposed to be more e cient, will be proposed, like pressure computations in SU(N) Yang-Mills theory simulations. The paperwork is structured as follows: Chapter 1 will brie y introduce the reader to Non Abelian gauge theories and then to Lattice implementation of such theories. Chapter 2 is dedicated to the QCD thermodynamics in the presence of a magnetic background. In Chapter 3 the numerical setup adopted for the implementation of the magnetic background it’s explained, and free energy measurements techniques developed so far in LQCD simulations involving magnetic backgrounds will be discussed. Finally, Chapter 4 deals with properties and issues regarding the algorithm based on Jarzynski relation. Finally, Chapter 5 and 6 are dedicated to the discussion of results, conclusions and perspectives.