You may be familiar with Kiefer’s The Carb Nite Solution: The Physicist’s Guide to Power Dieting. If you’re not, all you need to know is that it allows you to lose fat without counting calories. How is that possible? Why do low-carb diets work? And why does eating to satiety help, even when calories aren’t explicitly restricted?
In  the authors state that:
The first law of thermodynamics dictates that body mass remains constant when caloric intake equals caloric expenditure. It should be noted, however, that different diets lead to different biochemical pathways that are not equivalent when correctly compared through the laws of thermodynamics. It is inappropriate to assume that the only thing that counts in terms of food consumption and energy balance is the intake of dietary calories and weight storage. Well-controlled studies suggest that calorie content may not be as predictive of fat loss as is reduced carbohydrate consumption. Biologically speaking, a calorie is certainly not a calorie.
This is further supported by a literature review :
The principle of “a calorie is a calorie,” that weight change in hypocaloric diets is independent of macronutrient composition, is widely held in the popular and technical literature, and is frequently justified by appeal to the laws of thermodynamics. We review here some aspects of thermodynamics that bear on weight loss and the effect of macronutrient composition. The focus is the so-called metabolic advantage in low-carbohydrate diets – greater weight loss compared to isocaloric diets of different composition. Two laws of thermodynamics are relevant to the systems considered in nutrition and, whereas the first law is a conservation (of energy) law, the second is a dissipation law: something (negative entropy) is lost and therefore balance is not to be expected in diet interventions. Here, we propose that a misunderstanding of the second law accounts for the controversy about the role of macronutrient effect on weight loss and we review some aspects of elementary thermodynamics. We use data in the literature to show that thermogenesis is sufficient to predict metabolic advantage. Whereas homeostasis ensures balance under many conditions, as a general principle, “a calorie is a calorie” violates the second law of thermodynamics.
Returning to  again, there is a detailed description of what happens when carbohydrates are restricted:
The hormonal changes associated with a low-carbohydrate diet include a reduction in the circulating levels of insulin along with increased levels of glucagons, leading to activation of phosphoenolpyruvate carboxykinase, fructose 1,6-biphosphatase, and glucose 6-phosphatase and inhibition of pyruvate kinase, 6-phosphofructo-1-kinase, and hexokinase, favoring gluconeogenesis over glycolysis.
(Glycolysis is an energy pathway. It’s the process that burns carbohydrates for energy by oxidising glucose. Gluconeogenesis is the metabolic pathway that converts generates glucose from other substrates, like glucogenic amino acids.)
Gluconeogenesis is an energy-consuming process as 6 mol of ATP are consumed for the synthesis of 1 mol of glucose from pyruvate or lactate . The transformation of gluconeogenic amino acids into glucose requires even more energy because ATP is needed to dispose of the nitrogen as urea . Further, a low-carbohydrate diet increases turnover of body proteins; and the energy-dependent processes of maintaining the turnover of proteins (Table (Table1),1), including synthesis, folding, targeting, regulatory processes, and protein breakdown have an overall cost to body energy homeostasis that is significantly higher than previously appreciated .
Also, a low-carbohydrate diet is often high in protein. A recent study demonstrated that postprandial thermogenesis was increased 100% on a high-protein/low-fat diet vs. a high-carbohydrate/low-fat diet in healthy subjects .
A high-protein diet also spares muscle glycogen, lowers insulin, and decreases lipogenesis . Lower insulin levels increase catecholamine levels, which increases fat mobilisation.
Finally, ketogenic diets are characterized by elevations of free fatty acids, leading to the increased transcription of mitochondrial uncoupling proteins and of peroxisomal β-oxidation . Uncoupling proteins allow the proton gradient generated by the respiratory chain to re-enter the mitochondria by pathways which bypass the F1 ATPase, resulting in the generation of heat rather than ATP. Also, fatty acids undergoing β-oxidation with peroxisomes have no mechanism for energy conservation and result solely in heat production . The important bottom line is that this leads to metabolic inefficiency.
Interestingly, an increase in dietary fat intake also increases fat oxidation! Even better, it works for both lean and obese people .
Carbohydrate Restriction Mimics Fasting
You are probably familiar with the idea of intermittent fasting for fat loss. It turns out that you can get the same metabolic benefits simply by restricting carbohydrates . That doesn’t mean you should ditch fasting, skipping breakfast is still optimal, but don’t assume that extended fasts are necessary.
Hormones, Gene Expression and Transcription
PGC1-alpha is a mediator of many of the known beneficial effects of exercise . It decreases muscle atrophy, increases mitochondrial biogenesis, decreases apoptosis and protein breakdown, and promotes balanced autophagy (which is also necessary to prevent atrophy). It activates uncoupling protein 1 (UCP1), increasing thermogenesis and producing more brown adipose tissue. Additionally, it has a neuroprotective effect and may be cardioprotective too. Since PGC1-alpha activity decreases with age, and it is one of the key mechanisms by which calorie restriction (and fasting) increases lifespan, it may be vital for anti-aging and preventing age-related disease.
In  there were profound metabolic adaptations to a high fat diet, after only two days, with increased fat oxidation in both lean and obese subjects. In fact, the high fat groups had increased levels of PCG1-alpha, which implies that a high fat diet gives the same benefits as exercise!
Glucagon-like peptide 1 is a satiety hormone, but it has profound metabolic effects. The GLP-1 response after the ingestion of a meal may contribute significantly to fat loss. It was found in  that higher fasting plasma levels of GLP-1 were associated with higher resting energy expenditure and fat oxidation rates. It is likely that continued, elevated levels of GLP-1 are therefore desirable, and knowing that “a calorie is not a calorie,” as summarised above, this seems like a good explanation for why Carb Nite works so well.
Not only is there an increased oxidative capacity in skeletal muscle with a high fat diet , you can use fat for fuel during weight training, contrary to popular belief. The results from  showed that
Lipolysis may, therefore, provide energy during heavy-resistance exercise of relatively short duration. Also, storage and utilization of intramuscular substrates appear to be influenced by the metabolic profile of muscle.
and according to 
We conclude that intramyocellular lipid and glycogen stores are readily used during resistance exercise and this is likely associated with the reported increase in whole-body insulin sensitivity following resistance exercise.
This evidence is supported by another study  that showed:
In moderately active adult females, ingestion of a single LC meal resulted in greater lipid oxidation at rest and during exercise as compared to a single LF meal.